integral theorems · 2010. 9. 15. · logo1 antiderivativescauchy-goursat theoremtwo kinds of...
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![Page 1: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/1.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Integral Theorems
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 2: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/2.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction
1. Much of the strength of complex analysis derives from the factthat the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contourand in the contour.
2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit
more carefully.4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 3: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/3.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closedcontour is zero
, as long as the function is analytic on the contourand in the contour.
2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit
more carefully.4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 4: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/4.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contour
and in the contour.2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit
more carefully.4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 5: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/5.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contourand in the contour.
2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit
more carefully.4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 6: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/6.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contourand in the contour.
2. The above is called the Cauchy-Goursat Theorem.
3. We will start by analyzing integrals across closed contours a bitmore carefully.
4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 7: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/7.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contourand in the contour.
2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit
more carefully.
4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 8: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/8.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contourand in the contour.
2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit
more carefully.4. Then we will prove the Cauchy-Goursat Theorem.
5. Then we will consider a few properties of domains that relate tothe Cauchy-Goursat Theorem.
6. The original motivation to investigate integrals over closedcontours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 9: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/9.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contourand in the contour.
2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit
more carefully.4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.
6. The original motivation to investigate integrals over closedcontours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 10: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/10.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contourand in the contour.
2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit
more carefully.4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials inphysics.
For potentials in physics, integrals over closed curvesmust be zero.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 11: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/11.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closedcontour is zero, as long as the function is analytic on the contourand in the contour.
2. The above is called the Cauchy-Goursat Theorem.3. We will start by analyzing integrals across closed contours a bit
more carefully.4. Then we will prove the Cauchy-Goursat Theorem.5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials inphysics. For potentials in physics, integrals over closed curvesmust be zero.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 12: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/12.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem.
Let f be a continuous complex function on a domain D.The following are equivalent.
1. The function f has an antiderivative F on D. (That is,F′(z) = f (z) for all z in D.)
2. The integrals of f over any contour C from z1 to z2 in D onlydepends on z1 and z2, but not on C itself.
3. For any closed contour in D we have that∫
Cf (z) dz = 0.
In the above situation, if C is a contour from z1 to z2, then∫C
f (z) dz = F(z2)−F(z1).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 13: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/13.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let f be a continuous complex function on a domain D.
The following are equivalent.
1. The function f has an antiderivative F on D. (That is,F′(z) = f (z) for all z in D.)
2. The integrals of f over any contour C from z1 to z2 in D onlydepends on z1 and z2, but not on C itself.
3. For any closed contour in D we have that∫
Cf (z) dz = 0.
In the above situation, if C is a contour from z1 to z2, then∫C
f (z) dz = F(z2)−F(z1).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 14: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/14.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let f be a continuous complex function on a domain D.The following are equivalent.
1. The function f has an antiderivative F on D. (That is,F′(z) = f (z) for all z in D.)
2. The integrals of f over any contour C from z1 to z2 in D onlydepends on z1 and z2, but not on C itself.
3. For any closed contour in D we have that∫
Cf (z) dz = 0.
In the above situation, if C is a contour from z1 to z2, then∫C
f (z) dz = F(z2)−F(z1).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 15: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/15.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let f be a continuous complex function on a domain D.The following are equivalent.
1. The function f has an antiderivative F on D.
(That is,F′(z) = f (z) for all z in D.)
2. The integrals of f over any contour C from z1 to z2 in D onlydepends on z1 and z2, but not on C itself.
3. For any closed contour in D we have that∫
Cf (z) dz = 0.
In the above situation, if C is a contour from z1 to z2, then∫C
f (z) dz = F(z2)−F(z1).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 16: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/16.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let f be a continuous complex function on a domain D.The following are equivalent.
1. The function f has an antiderivative F on D. (That is,F′(z) = f (z) for all z in D.)
2. The integrals of f over any contour C from z1 to z2 in D onlydepends on z1 and z2, but not on C itself.
3. For any closed contour in D we have that∫
Cf (z) dz = 0.
In the above situation, if C is a contour from z1 to z2, then∫C
f (z) dz = F(z2)−F(z1).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 17: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/17.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let f be a continuous complex function on a domain D.The following are equivalent.
1. The function f has an antiderivative F on D. (That is,F′(z) = f (z) for all z in D.)
2. The integrals of f over any contour C from z1 to z2 in D onlydepends on z1 and z2, but not on C itself.
3. For any closed contour in D we have that∫
Cf (z) dz = 0.
In the above situation, if C is a contour from z1 to z2, then∫C
f (z) dz = F(z2)−F(z1).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 18: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/18.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let f be a continuous complex function on a domain D.The following are equivalent.
1. The function f has an antiderivative F on D. (That is,F′(z) = f (z) for all z in D.)
2. The integrals of f over any contour C from z1 to z2 in D onlydepends on z1 and z2, but not on C itself.
3. For any closed contour in D we have that∫
Cf (z) dz = 0.
In the above situation, if C is a contour from z1 to z2, then∫C
f (z) dz = F(z2)−F(z1).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 19: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/19.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let f be a continuous complex function on a domain D.The following are equivalent.
1. The function f has an antiderivative F on D. (That is,F′(z) = f (z) for all z in D.)
2. The integrals of f over any contour C from z1 to z2 in D onlydepends on z1 and z2, but not on C itself.
3. For any closed contour in D we have that∫
Cf (z) dz = 0.
In the above situation, if C is a contour from z1 to z2, then∫C
f (z) dz = F(z2)−F(z1).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 20: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/20.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example.
The integral of any power function zn with n 6=−1 beingan integer around any closed contour in the domain of the powerfunction is 0.
An antiderivative of zn is1
n+1zn+1!
(Also see earlier presentation for the direct computation for the unitcircle.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 21: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/21.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of any power function zn with n 6=−1 beingan integer around any closed contour in the domain of the powerfunction is 0.
An antiderivative of zn is1
n+1zn+1!
(Also see earlier presentation for the direct computation for the unitcircle.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 22: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/22.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of any power function zn with n 6=−1 beingan integer around any closed contour in the domain of the powerfunction is 0.
An antiderivative of zn is1
n+1zn+1!
(Also see earlier presentation for the direct computation for the unitcircle.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 23: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/23.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of any power function zn with n 6=−1 beingan integer around any closed contour in the domain of the powerfunction is 0.
An antiderivative of zn is1
n+1zn+1!
(Also see earlier presentation for the direct computation for the unitcircle.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 24: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/24.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example.
The integral of the function z−1 around the unit circle is2πi(?)
It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫
Cz−1 dz =
∫ 2π
0
(eit)−1
ieit dt
=∫ 2π
0i dt = 2πi
The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 25: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/25.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of the function z−1 around the unit circle is2πi
(?)
It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫
Cz−1 dz =
∫ 2π
0
(eit)−1
ieit dt
=∫ 2π
0i dt = 2πi
The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 26: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/26.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of the function z−1 around the unit circle is2πi(?)
It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫
Cz−1 dz =
∫ 2π
0
(eit)−1
ieit dt
=∫ 2π
0i dt = 2πi
The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 27: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/27.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of the function z−1 around the unit circle is2πi(?)
It’s tempting to declare the logarithm an antiderivative of z−1.
Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫
Cz−1 dz =
∫ 2π
0
(eit)−1
ieit dt
=∫ 2π
0i dt = 2πi
The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 28: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/28.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of the function z−1 around the unit circle is2πi(?)
It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero
(it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫
Cz−1 dz =
∫ 2π
0
(eit)−1
ieit dt
=∫ 2π
0i dt = 2πi
The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 29: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/29.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of the function z−1 around the unit circle is2πi(?)
It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible
, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫
Cz−1 dz =
∫ 2π
0
(eit)−1
ieit dt
=∫ 2π
0i dt = 2πi
The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 30: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/30.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of the function z−1 around the unit circle is2πi(?)
It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches)
, the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫
Cz−1 dz =
∫ 2π
0
(eit)−1
ieit dt
=∫ 2π
0i dt = 2πi
The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 31: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/31.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of the function z−1 around the unit circle is2πi(?)
It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.
Also recall ∫C
z−1 dz =∫ 2π
0
(eit)−1
ieit dt
=∫ 2π
0i dt = 2πi
The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 32: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/32.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of the function z−1 around the unit circle is2πi(?)
It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall
∫C
z−1 dz =∫ 2π
0
(eit)−1
ieit dt
=∫ 2π
0i dt = 2πi
The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 33: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/33.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of the function z−1 around the unit circle is2πi(?)
It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫
Cz−1 dz
=∫ 2π
0
(eit)−1
ieit dt
=∫ 2π
0i dt = 2πi
The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 34: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/34.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of the function z−1 around the unit circle is2πi(?)
It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫
Cz−1 dz =
∫ 2π
0
(eit)−1
ieit dt
=∫ 2π
0i dt = 2πi
The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 35: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/35.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of the function z−1 around the unit circle is2πi(?)
It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫
Cz−1 dz =
∫ 2π
0
(eit)−1
ieit dt
=∫ 2π
0i dt
= 2πi
The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 36: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/36.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of the function z−1 around the unit circle is2πi(?)
It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫
Cz−1 dz =
∫ 2π
0
(eit)−1
ieit dt
=∫ 2π
0i dt = 2πi
The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 37: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/37.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of the function z−1 around the unit circle is2πi(?)
It’s tempting to declare the logarithm an antiderivative of z−1. Butbecause of the problems with defining a logarithm function on adeleted neighborhood of zero (it’s not possible, that’s why we workwith branches), the logarithm is not an antiderivative of z−1 on anydeleted neighborhood of zero.Also recall ∫
Cz−1 dz =
∫ 2π
0
(eit)−1
ieit dt
=∫ 2π
0i dt = 2πi
The logarithm is an antiderivative of z−1 on any subset of the complexnumbers from which an appropriate branch cut has been removed,though.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 38: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/38.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“1⇒3”).
If F is an antiderivative of f and C is a closedcontour from z(a) to z(b) = z(a), then∫
Cf (z) dz = F
(z(b)
)−F(z(a)
)= F
(z(a)
)−F(z(a)
)= 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 39: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/39.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“1⇒3”). If F is an antiderivative of f and C is a closedcontour from z(a) to z(b) = z(a), then∫
Cf (z) dz
= F(z(b)
)−F(z(a)
)= F
(z(a)
)−F(z(a)
)= 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 40: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/40.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“1⇒3”). If F is an antiderivative of f and C is a closedcontour from z(a) to z(b) = z(a), then∫
Cf (z) dz = F
(z(b)
)−F(z(a)
)
= F(z(a)
)−F(z(a)
)= 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 41: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/41.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“1⇒3”). If F is an antiderivative of f and C is a closedcontour from z(a) to z(b) = z(a), then∫
Cf (z) dz = F
(z(b)
)−F(z(a)
)= F
(z(a)
)−F(z(a)
)
= 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 42: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/42.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“1⇒3”). If F is an antiderivative of f and C is a closedcontour from z(a) to z(b) = z(a), then∫
Cf (z) dz = F
(z(b)
)−F(z(a)
)= F
(z(a)
)−F(z(a)
)= 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 43: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/43.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“3⇒2”).
Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.
0 =∫
Cf (z) dz
=∫
C1+(−C2)f (z) dz
=∫
C1f (z) dz−
∫C2
f (z) dz∫C1
f (z) dz =∫
C2f (z) dz
Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 44: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/44.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2).
Note thatC := C1 +(−C2) is a closed contour.
0 =∫
Cf (z) dz
=∫
C1+(−C2)f (z) dz
=∫
C1f (z) dz−
∫C2
f (z) dz∫C1
f (z) dz =∫
C2f (z) dz
Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 45: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/45.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.
0 =∫
Cf (z) dz
=∫
C1+(−C2)f (z) dz
=∫
C1f (z) dz−
∫C2
f (z) dz∫C1
f (z) dz =∫
C2f (z) dz
Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 46: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/46.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.
0 =∫
Cf (z) dz
=∫
C1+(−C2)f (z) dz
=∫
C1f (z) dz−
∫C2
f (z) dz∫C1
f (z) dz =∫
C2f (z) dz
Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 47: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/47.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.
0 =∫
Cf (z) dz
=∫
C1+(−C2)f (z) dz
=∫
C1f (z) dz−
∫C2
f (z) dz∫C1
f (z) dz =∫
C2f (z) dz
Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 48: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/48.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.
0 =∫
Cf (z) dz
=∫
C1+(−C2)f (z) dz
=∫
C1f (z) dz
−∫
C2f (z) dz∫
C1f (z) dz =
∫C2
f (z) dz
Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 49: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/49.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.
0 =∫
Cf (z) dz
=∫
C1+(−C2)f (z) dz
=∫
C1f (z) dz−
∫C2
f (z) dz
∫C1
f (z) dz =∫
C2f (z) dz
Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 50: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/50.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.
0 =∫
Cf (z) dz
=∫
C1+(−C2)f (z) dz
=∫
C1f (z) dz−
∫C2
f (z) dz∫C1
f (z) dz =∫
C2f (z) dz
Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 51: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/51.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.
0 =∫
Cf (z) dz
=∫
C1+(−C2)f (z) dz
=∫
C1f (z) dz−
∫C2
f (z) dz∫C1
f (z) dz =∫
C2f (z) dz
Thus the integrals along any contour from z1 to z2 all have the samevalue
, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 52: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/52.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.
0 =∫
Cf (z) dz
=∫
C1+(−C2)f (z) dz
=∫
C1f (z) dz−
∫C2
f (z) dz∫C1
f (z) dz =∫
C2f (z) dz
Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2
, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 53: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/53.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.
0 =∫
Cf (z) dz
=∫
C1+(−C2)f (z) dz
=∫
C1f (z) dz−
∫C2
f (z) dz∫C1
f (z) dz =∫
C2f (z) dz
Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2
, not on the path we take from one point to theother.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 54: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/54.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“3⇒2”). Let C1 and C2 be two contours fromz1 = z1(a1) = z2(a2) to z2 = z1(b1) = z2(b2). Note thatC := C1 +(−C2) is a closed contour.
0 =∫
Cf (z) dz
=∫
C1+(−C2)f (z) dz
=∫
C1f (z) dz−
∫C2
f (z) dz∫C1
f (z) dz =∫
C2f (z) dz
Thus the integrals along any contour from z1 to z2 all have the samevalue, which means that, for arbitrary z1 and z2, the integral onlydepends on z1 and z2, not on the path we take from one point to theother.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 55: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/55.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“2⇒1”).
Fix a point z0 in D. For z in D (recall that D is
connected) define F(z) :=∫ z
z0
f (γ) dγ . Because the integral only
depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.
limw→z
F(w)−F(z)w− z
= limw→z
1w− z
(∫ w
z0
f (γ) dγ−∫ z
z0
f (γ) dγ
)= lim
w→z
1w− z
∫ w
zf (γ) dγ = f (z)
because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 56: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/56.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“2⇒1”). Fix a point z0 in D.
For z in D (recall that D is
connected) define F(z) :=∫ z
z0
f (γ) dγ . Because the integral only
depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.
limw→z
F(w)−F(z)w− z
= limw→z
1w− z
(∫ w
z0
f (γ) dγ−∫ z
z0
f (γ) dγ
)= lim
w→z
1w− z
∫ w
zf (γ) dγ = f (z)
because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 57: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/57.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“2⇒1”). Fix a point z0 in D. For z in D
(recall that D is
connected) define F(z) :=∫ z
z0
f (γ) dγ . Because the integral only
depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.
limw→z
F(w)−F(z)w− z
= limw→z
1w− z
(∫ w
z0
f (γ) dγ−∫ z
z0
f (γ) dγ
)= lim
w→z
1w− z
∫ w
zf (γ) dγ = f (z)
because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 58: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/58.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is
connected)
define F(z) :=∫ z
z0
f (γ) dγ . Because the integral only
depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.
limw→z
F(w)−F(z)w− z
= limw→z
1w− z
(∫ w
z0
f (γ) dγ−∫ z
z0
f (γ) dγ
)= lim
w→z
1w− z
∫ w
zf (γ) dγ = f (z)
because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 59: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/59.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is
connected) define F(z) :=∫ z
z0
f (γ) dγ .
Because the integral only
depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.
limw→z
F(w)−F(z)w− z
= limw→z
1w− z
(∫ w
z0
f (γ) dγ−∫ z
z0
f (γ) dγ
)= lim
w→z
1w− z
∫ w
zf (γ) dγ = f (z)
because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 60: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/60.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is
connected) define F(z) :=∫ z
z0
f (γ) dγ . Because the integral only
depends on the endpoints, we need not specify the contour
, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.
limw→z
F(w)−F(z)w− z
= limw→z
1w− z
(∫ w
z0
f (γ) dγ−∫ z
z0
f (γ) dγ
)= lim
w→z
1w− z
∫ w
zf (γ) dγ = f (z)
because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 61: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/61.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is
connected) define F(z) :=∫ z
z0
f (γ) dγ . Because the integral only
depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line.
We claim that F′ = f . Let z be in D.
limw→z
F(w)−F(z)w− z
= limw→z
1w− z
(∫ w
z0
f (γ) dγ−∫ z
z0
f (γ) dγ
)= lim
w→z
1w− z
∫ w
zf (γ) dγ = f (z)
because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 62: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/62.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is
connected) define F(z) :=∫ z
z0
f (γ) dγ . Because the integral only
depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f .
Let z be in D.
limw→z
F(w)−F(z)w− z
= limw→z
1w− z
(∫ w
z0
f (γ) dγ−∫ z
z0
f (γ) dγ
)= lim
w→z
1w− z
∫ w
zf (γ) dγ = f (z)
because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 63: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/63.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is
connected) define F(z) :=∫ z
z0
f (γ) dγ . Because the integral only
depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.
limw→z
F(w)−F(z)w− z
= limw→z
1w− z
(∫ w
z0
f (γ) dγ−∫ z
z0
f (γ) dγ
)= lim
w→z
1w− z
∫ w
zf (γ) dγ = f (z)
because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 64: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/64.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is
connected) define F(z) :=∫ z
z0
f (γ) dγ . Because the integral only
depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.
limw→z
F(w)−F(z)w− z
= limw→z
1w− z
(∫ w
z0
f (γ) dγ−∫ z
z0
f (γ) dγ
)= lim
w→z
1w− z
∫ w
zf (γ) dγ = f (z)
because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 65: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/65.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is
connected) define F(z) :=∫ z
z0
f (γ) dγ . Because the integral only
depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.
limw→z
F(w)−F(z)w− z
= limw→z
1w− z
(∫ w
z0
f (γ) dγ−∫ z
z0
f (γ) dγ
)
= limw→z
1w− z
∫ w
zf (γ) dγ = f (z)
because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 66: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/66.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is
connected) define F(z) :=∫ z
z0
f (γ) dγ . Because the integral only
depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.
limw→z
F(w)−F(z)w− z
= limw→z
1w− z
(∫ w
z0
f (γ) dγ−∫ z
z0
f (γ) dγ
)= lim
w→z
1w− z
∫ w
zf (γ) dγ
= f (z)
because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 67: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/67.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is
connected) define F(z) :=∫ z
z0
f (γ) dγ . Because the integral only
depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.
limw→z
F(w)−F(z)w− z
= limw→z
1w− z
(∫ w
z0
f (γ) dγ−∫ z
z0
f (γ) dγ
)= lim
w→z
1w− z
∫ w
zf (γ) dγ = f (z)
because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 68: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/68.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is
connected) define F(z) :=∫ z
z0
f (γ) dγ . Because the integral only
depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.
limw→z
F(w)−F(z)w− z
= limw→z
1w− z
(∫ w
z0
f (γ) dγ−∫ z
z0
f (γ) dγ
)= lim
w→z
1w− z
∫ w
zf (γ) dγ = f (z)
because, with the contour from z to w chosen to be a straight line
, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 69: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/69.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is
connected) define F(z) :=∫ z
z0
f (γ) dγ . Because the integral only
depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.
limw→z
F(w)−F(z)w− z
= limw→z
1w− z
(∫ w
z0
f (γ) dγ−∫ z
z0
f (γ) dγ
)= lim
w→z
1w− z
∫ w
zf (γ) dγ = f (z)
because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z)
, so that the integral is close(“and in the limit equal”) to f (z)(w− z).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 70: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/70.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (“2⇒1”). Fix a point z0 in D. For z in D (recall that D is
connected) define F(z) :=∫ z
z0
f (γ) dγ . Because the integral only
depends on the endpoints, we need not specify the contour, and in thiscase it is common to use notation that is similar to that for integralsover intervals on the real line. We claim that F′ = f . Let z be in D.
limw→z
F(w)−F(z)w− z
= limw→z
1w− z
(∫ w
z0
f (γ) dγ−∫ z
z0
f (γ) dγ
)= lim
w→z
1w− z
∫ w
zf (γ) dγ = f (z)
because, with the contour from z to w chosen to be a straight line, asw→ z, the values f (γ) are close to f (z), so that the integral is close(“and in the limit equal”) to f (z)(w− z).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 71: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/71.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (finish).
Finally, the claimed equation follows from a resultfrom a previous presentation.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 72: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/72.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (finish). Finally, the claimed equation follows from a resultfrom a previous presentation.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 73: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/73.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof (finish). Finally, the claimed equation follows from a resultfrom a previous presentation.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 74: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/74.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem.
Let C be a simple closed contour and let f be a complexfunction that is analytic on a domain that contains C and the interior
of C. Then∫
Cf (z) dz = 0.
C
interior
exterior
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 75: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/75.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C be a simple closed contour and let f be a complexfunction that is analytic on a domain that contains C and the interior
of C.
Then∫
Cf (z) dz = 0.
C
interior
exterior
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 76: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/76.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C be a simple closed contour and let f be a complexfunction that is analytic on a domain that contains C and the interior
of C. Then∫
Cf (z) dz = 0.
C
interior
exterior
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 77: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/77.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C be a simple closed contour and let f be a complexfunction that is analytic on a domain that contains C and the interior
of C. Then∫
Cf (z) dz = 0.
C
interior
exterior
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 78: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/78.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C be a simple closed contour and let f be a complexfunction that is analytic on a domain that contains C and the interior
of C. Then∫
Cf (z) dz = 0.
C
interior
exterior
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 79: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/79.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C be a simple closed contour and let f be a complexfunction that is analytic on a domain that contains C and the interior
of C. Then∫
Cf (z) dz = 0.
C
interior
exterior
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 80: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/80.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 81: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/81.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c.
Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 82: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/82.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a.
Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 83: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/83.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2
, mbc :=b+ c
2, and mca :=
c+a2
. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 84: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/84.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 85: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/85.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2.
Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 86: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/86.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 87: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/87.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 88: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/88.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 89: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/89.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 90: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/90.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 91: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/91.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 92: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/92.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 93: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/93.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
r
mab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 94: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/94.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 95: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/95.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
r
mbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 96: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/96.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbc
rmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 97: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/97.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcr
mca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 98: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/98.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 99: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/99.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 100: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/100.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 101: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/101.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 102: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/102.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz
=∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 103: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/103.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz
+∫
[b,mbc,mab,b]f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 104: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/104.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 105: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/105.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz
+∫
[mab,mbc,mca,mab]f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 106: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/106.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. For three complex numbers a,b,c let ∆(a,b,c) be the trianglewith vertices a,b,c. Let [a,b,c,a] denote the curve that traverses thesides of the triangle from a to b to c and back to a. Denote the
midpoints mab :=a+b
2, mbc :=
b+ c2
, and mca :=c+a
2. Then
ra
rb
rc
rmab
rmbcrmca
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 107: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/107.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz
implies that the absolute value of one of the integrals on the right is
greater than or equal to∣∣∣∣14∫
[a,b,c,a]f (z) dz
∣∣∣∣ . Thus, if a0 := a, b0 := b,
c0 := c, then we can find a1,b1,c1 so that∣∣∣∣∫[a1,b1,c1,a1]f (z) dz
∣∣∣∣≥ 14
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣so that the lengths of the sides satisfy
l([a1,b1,c1,a1]
)≤ 1
2l([a0,b0,c0,a0]
)and so that the diameters satisfy
diam(∆(a1,b1,c1)
)≤ 1
2diam
(∆(a0,b0,c0)
).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 108: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/108.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz
implies that the absolute value of one of the integrals on the right is
greater than or equal to∣∣∣∣14∫
[a,b,c,a]f (z) dz
∣∣∣∣ . Thus, if a0 := a, b0 := b,
c0 := c, then we can find a1,b1,c1 so that∣∣∣∣∫[a1,b1,c1,a1]f (z) dz
∣∣∣∣≥ 14
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣so that the lengths of the sides satisfy
l([a1,b1,c1,a1]
)≤ 1
2l([a0,b0,c0,a0]
)and so that the diameters satisfy
diam(∆(a1,b1,c1)
)≤ 1
2diam
(∆(a0,b0,c0)
).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 109: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/109.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz
implies that the absolute value of one of the integrals on the right is
greater than or equal to∣∣∣∣14∫
[a,b,c,a]f (z) dz
∣∣∣∣ . Thus, if a0 := a
, b0 := b,
c0 := c, then we can find a1,b1,c1 so that∣∣∣∣∫[a1,b1,c1,a1]f (z) dz
∣∣∣∣≥ 14
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣so that the lengths of the sides satisfy
l([a1,b1,c1,a1]
)≤ 1
2l([a0,b0,c0,a0]
)and so that the diameters satisfy
diam(∆(a1,b1,c1)
)≤ 1
2diam
(∆(a0,b0,c0)
).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 110: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/110.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz
implies that the absolute value of one of the integrals on the right is
greater than or equal to∣∣∣∣14∫
[a,b,c,a]f (z) dz
∣∣∣∣ . Thus, if a0 := a, b0 := b
,
c0 := c, then we can find a1,b1,c1 so that∣∣∣∣∫[a1,b1,c1,a1]f (z) dz
∣∣∣∣≥ 14
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣so that the lengths of the sides satisfy
l([a1,b1,c1,a1]
)≤ 1
2l([a0,b0,c0,a0]
)and so that the diameters satisfy
diam(∆(a1,b1,c1)
)≤ 1
2diam
(∆(a0,b0,c0)
).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 111: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/111.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz
implies that the absolute value of one of the integrals on the right is
greater than or equal to∣∣∣∣14∫
[a,b,c,a]f (z) dz
∣∣∣∣ . Thus, if a0 := a, b0 := b,
c0 := c
, then we can find a1,b1,c1 so that∣∣∣∣∫[a1,b1,c1,a1]f (z) dz
∣∣∣∣≥ 14
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣so that the lengths of the sides satisfy
l([a1,b1,c1,a1]
)≤ 1
2l([a0,b0,c0,a0]
)and so that the diameters satisfy
diam(∆(a1,b1,c1)
)≤ 1
2diam
(∆(a0,b0,c0)
).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 112: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/112.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz
implies that the absolute value of one of the integrals on the right is
greater than or equal to∣∣∣∣14∫
[a,b,c,a]f (z) dz
∣∣∣∣ . Thus, if a0 := a, b0 := b,
c0 := c, then we can find a1,b1,c1 so that∣∣∣∣∫[a1,b1,c1,a1]f (z) dz
∣∣∣∣≥ 14
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣
so that the lengths of the sides satisfy
l([a1,b1,c1,a1]
)≤ 1
2l([a0,b0,c0,a0]
)and so that the diameters satisfy
diam(∆(a1,b1,c1)
)≤ 1
2diam
(∆(a0,b0,c0)
).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 113: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/113.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz
implies that the absolute value of one of the integrals on the right is
greater than or equal to∣∣∣∣14∫
[a,b,c,a]f (z) dz
∣∣∣∣ . Thus, if a0 := a, b0 := b,
c0 := c, then we can find a1,b1,c1 so that∣∣∣∣∫[a1,b1,c1,a1]f (z) dz
∣∣∣∣≥ 14
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣so that the lengths of the sides satisfy
l([a1,b1,c1,a1]
)≤ 1
2l([a0,b0,c0,a0]
)
and so that the diameters satisfy
diam(∆(a1,b1,c1)
)≤ 1
2diam
(∆(a0,b0,c0)
).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 114: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/114.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.∫[a,b,c,a]
f (z) dz =∫
[a,mab,mca,a]f (z) dz+
∫[b,mbc,mab,b]
f (z) dz
+∫
[c,mca,mbc,b]f (z) dz+
∫[mab,mbc,mca,mab]
f (z) dz
implies that the absolute value of one of the integrals on the right is
greater than or equal to∣∣∣∣14∫
[a,b,c,a]f (z) dz
∣∣∣∣ . Thus, if a0 := a, b0 := b,
c0 := c, then we can find a1,b1,c1 so that∣∣∣∣∫[a1,b1,c1,a1]f (z) dz
∣∣∣∣≥ 14
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣so that the lengths of the sides satisfy
l([a1,b1,c1,a1]
)≤ 1
2l([a0,b0,c0,a0]
)and so that the diameters satisfy
diam(∆(a1,b1,c1)
)≤ 1
2diam
(∆(a0,b0,c0)
).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 115: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/115.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 116: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/116.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 117: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/117.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 118: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/118.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 119: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/119.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 120: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/120.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 121: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/121.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 122: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/122.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 123: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/123.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1
rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 124: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/124.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 125: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/125.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 126: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/126.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 127: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/127.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 128: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/128.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 129: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/129.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2
rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 130: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/130.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2
qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 131: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/131.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2
qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 132: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/132.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2
qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 133: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/133.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2
qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 134: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/134.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 135: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/135.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 136: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/136.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3
qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 137: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/137.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3
qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 138: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/138.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3
qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 139: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/139.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3
qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 140: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/140.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3q
qq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 141: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/141.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qq
q∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 142: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/142.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq
∣∣∣∣∫[an,bn,cn,an]f (z) dz
∣∣∣∣ ≥ 14
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 143: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/143.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq
∣∣∣∣∫[an,bn,cn,an]f (z) dz
∣∣∣∣ ≥ 14
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 144: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/144.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq
∣∣∣∣∫[an,bn,cn,an]f (z) dz
∣∣∣∣ ≥ 14
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 145: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/145.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq
∣∣∣∣∫[an,bn,cn,an]f (z) dz
∣∣∣∣ ≥ 14
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 146: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/146.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq
∣∣∣∣∫[an,bn,cn,an]f (z) dz
∣∣∣∣ ≥ 14
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 147: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/147.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣
≥ ·· ·
≥ 14n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 148: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/148.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·
≥ 14n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 149: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/149.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣
l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 150: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/150.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)
≤ ·· · ≤ 12n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 151: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/151.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· ·
≤ 12n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 152: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/152.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)
diam(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 153: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/153.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)
≤·· · ≤ 12n diam
(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 154: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/154.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· ·
≤ 12n diam
(∆(a0,b0,c0)
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 155: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/155.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
ra0
rb0
rc0
ra1
rb1rc1
ra2
rb2rc2 qa3
qb3
qc3qqq∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣ ≥ 1
4
∣∣∣∣∫[an−1,bn−1,cn−1,an−1]f (z) dz
∣∣∣∣ ≥ ·· ·≥ 1
4n
∣∣∣∣∫[a0,b0,c0,a0]f (z) dz
∣∣∣∣l([an,bn,cn,an]
)≤ 1
2l([an−1,bn−1,cn−1,an−1]
)≤ ·· · ≤ 1
2n l([a0,b0,c0,a0]
)diam
(∆(an,bn,cn)
)≤ 1
2diam
(∆(an−1,bn−1,cn−1)
)≤·· · ≤ 1
2n diam(∆(a0,b0,c0)
)Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 156: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/156.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
By definition of the derivative
0 = limz→z0
f (z)− f (z0)z− z0
− f ′(z0) =: limz→z0
h(z)
Thus there is a function h so that for all z we have
f (z) = f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0)
and limz→z0
h(z) = 0. For any ε > 0 we can find an n so that
supz∈[an,bn,cn,an]
∣∣h(z)∣∣< ε
l[a,b,c,a]diam(∆(a,b,c)
) .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 157: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/157.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. By definition of the derivative
0 = limz→z0
f (z)− f (z0)z− z0
− f ′(z0)
=: limz→z0
h(z)
Thus there is a function h so that for all z we have
f (z) = f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0)
and limz→z0
h(z) = 0. For any ε > 0 we can find an n so that
supz∈[an,bn,cn,an]
∣∣h(z)∣∣< ε
l[a,b,c,a]diam(∆(a,b,c)
) .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 158: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/158.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. By definition of the derivative
0 = limz→z0
f (z)− f (z0)z− z0
− f ′(z0) =: limz→z0
h(z)
Thus there is a function h so that for all z we have
f (z) = f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0)
and limz→z0
h(z) = 0. For any ε > 0 we can find an n so that
supz∈[an,bn,cn,an]
∣∣h(z)∣∣< ε
l[a,b,c,a]diam(∆(a,b,c)
) .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 159: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/159.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. By definition of the derivative
0 = limz→z0
f (z)− f (z0)z− z0
− f ′(z0) =: limz→z0
h(z)
Thus there is a function h so that for all z we have
f (z) = f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0)
and limz→z0
h(z) = 0. For any ε > 0 we can find an n so that
supz∈[an,bn,cn,an]
∣∣h(z)∣∣< ε
l[a,b,c,a]diam(∆(a,b,c)
) .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 160: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/160.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. By definition of the derivative
0 = limz→z0
f (z)− f (z0)z− z0
− f ′(z0) =: limz→z0
h(z)
Thus there is a function h so that for all z we have
f (z) = f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0)
and limz→z0
h(z) = 0.
For any ε > 0 we can find an n so that
supz∈[an,bn,cn,an]
∣∣h(z)∣∣< ε
l[a,b,c,a]diam(∆(a,b,c)
) .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 161: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/161.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. By definition of the derivative
0 = limz→z0
f (z)− f (z0)z− z0
− f ′(z0) =: limz→z0
h(z)
Thus there is a function h so that for all z we have
f (z) = f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0)
and limz→z0
h(z) = 0. For any ε > 0 we can find an n so that
supz∈[an,bn,cn,an]
∣∣h(z)∣∣< ε
l[a,b,c,a]diam(∆(a,b,c)
) .
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 162: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/162.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
Then∣∣∣∣∫[a,b,c,a]f (z) dz
∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
h(z)(z− z0) dz∣∣∣∣ ≤ 4n
∫[an,bn,cn,an]
∣∣(z− z0)∣∣∣∣h(z)
∣∣ d|z|
≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
= 4n 12n l[a,b,c,a]
12n diam
(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
≤ l[a,b,c,a]diam(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣< ε
implies that∫
[a,b,c,a]f (z) dz = 0, because ε was arbitrary.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 163: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/163.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz
∣∣∣∣
≤ 4n∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
h(z)(z− z0) dz∣∣∣∣ ≤ 4n
∫[an,bn,cn,an]
∣∣(z− z0)∣∣∣∣h(z)
∣∣ d|z|
≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
= 4n 12n l[a,b,c,a]
12n diam
(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
≤ l[a,b,c,a]diam(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣< ε
implies that∫
[a,b,c,a]f (z) dz = 0, because ε was arbitrary.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 164: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/164.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz
∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
h(z)(z− z0) dz∣∣∣∣ ≤ 4n
∫[an,bn,cn,an]
∣∣(z− z0)∣∣∣∣h(z)
∣∣ d|z|
≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
= 4n 12n l[a,b,c,a]
12n diam
(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
≤ l[a,b,c,a]diam(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣< ε
implies that∫
[a,b,c,a]f (z) dz = 0, because ε was arbitrary.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 165: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/165.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz
∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
h(z)(z− z0) dz∣∣∣∣ ≤ 4n
∫[an,bn,cn,an]
∣∣(z− z0)∣∣∣∣h(z)
∣∣ d|z|
≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
= 4n 12n l[a,b,c,a]
12n diam
(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
≤ l[a,b,c,a]diam(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣< ε
implies that∫
[a,b,c,a]f (z) dz = 0, because ε was arbitrary.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 166: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/166.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz
∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
h(z)(z− z0) dz∣∣∣∣
≤ 4n∫
[an,bn,cn,an]
∣∣(z− z0)∣∣∣∣h(z)
∣∣ d|z|
≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
= 4n 12n l[a,b,c,a]
12n diam
(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
≤ l[a,b,c,a]diam(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣< ε
implies that∫
[a,b,c,a]f (z) dz = 0, because ε was arbitrary.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 167: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/167.jpg)
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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz
∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
h(z)(z− z0) dz∣∣∣∣ ≤ 4n
∫[an,bn,cn,an]
∣∣(z− z0)∣∣∣∣h(z)
∣∣ d|z|
≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
= 4n 12n l[a,b,c,a]
12n diam
(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
≤ l[a,b,c,a]diam(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣< ε
implies that∫
[a,b,c,a]f (z) dz = 0, because ε was arbitrary.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 168: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/168.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz
∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
h(z)(z− z0) dz∣∣∣∣ ≤ 4n
∫[an,bn,cn,an]
∣∣(z− z0)∣∣∣∣h(z)
∣∣ d|z|
≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
= 4n 12n l[a,b,c,a]
12n diam
(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
≤ l[a,b,c,a]diam(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣< ε
implies that∫
[a,b,c,a]f (z) dz = 0, because ε was arbitrary.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 169: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/169.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz
∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
h(z)(z− z0) dz∣∣∣∣ ≤ 4n
∫[an,bn,cn,an]
∣∣(z− z0)∣∣∣∣h(z)
∣∣ d|z|
≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
= 4n 12n l[a,b,c,a]
12n diam
(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
≤ l[a,b,c,a]diam(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣< ε
implies that∫
[a,b,c,a]f (z) dz = 0, because ε was arbitrary.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 170: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/170.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz
∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
h(z)(z− z0) dz∣∣∣∣ ≤ 4n
∫[an,bn,cn,an]
∣∣(z− z0)∣∣∣∣h(z)
∣∣ d|z|
≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
= 4n 12n l[a,b,c,a]
12n diam
(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
≤ l[a,b,c,a]diam(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣< ε
implies that∫
[a,b,c,a]f (z) dz = 0, because ε was arbitrary.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 171: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/171.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof. Then∣∣∣∣∫[a,b,c,a]f (z) dz
∣∣∣∣ ≤ 4n∣∣∣∣∫[an,bn,cn,an]
f (z) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
f (z0)+ f ′(z0)(z− z0)+h(z)(z− z0) dz∣∣∣∣
= 4n∣∣∣∣∫[an,bn,cn,an]
h(z)(z− z0) dz∣∣∣∣ ≤ 4n
∫[an,bn,cn,an]
∣∣(z− z0)∣∣∣∣h(z)
∣∣ d|z|
≤ 4nl[an,bn,cn,an]diam(∆(an,bn,cn)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
= 4n 12n l[a,b,c,a]
12n diam
(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣
≤ l[a,b,c,a]diam(∆(a,b,c)
)sup
z∈[an,bn,cn,an]
∣∣h(z)∣∣< ε
implies that∫
[a,b,c,a]f (z) dz = 0, because ε was arbitrary.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 172: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/172.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 173: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/173.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 174: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/174.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 175: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/175.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
t
t tt
t
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 176: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/176.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt
tt
t
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 177: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/177.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 178: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/178.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
t
t
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 179: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/179.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 180: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/180.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
t
ttt t t
ttt
t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 181: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/181.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
tt
tt t t
ttt
t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 182: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/182.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 183: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/183.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t
t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 184: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/184.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t
t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 185: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/185.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t
t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 186: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/186.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t
t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 187: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/187.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t
t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 188: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/188.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t
t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 189: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/189.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t
t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 190: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/190.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t
t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 191: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/191.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t
t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 192: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/192.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t
t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 193: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/193.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t
t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 194: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/194.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t
tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 195: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/195.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t t
ttt
t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 196: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/196.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 197: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/197.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
t
tt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 198: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/198.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
tt
t
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 199: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/199.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 200: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/200.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 201: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/201.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 202: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/202.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 203: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/203.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 204: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/204.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 205: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/205.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 206: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/206.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 207: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/207.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 208: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/208.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 209: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/209.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 210: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/210.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 211: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/211.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 212: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/212.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 213: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/213.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 214: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/214.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 215: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/215.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 216: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/216.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 217: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/217.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 218: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/218.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 219: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/219.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 220: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/220.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 221: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/221.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 222: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/222.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 223: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/223.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 224: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/224.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 225: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/225.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
C
t
tt t
tt
ttt
t t tt
ttt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 226: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/226.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Domains With and Without Holes
1. The Cauchy-Goursat Theorem works as long as the function isanalytic on a domain that contains the contour and the contour’sinterior.
2. But if C is the positively oriented unit circle, then∫C
z−1 dz = 2πi 6= 0 shows that the result need not hold when the
function is not analytic in the whole interior.
3. Side note:∫
Cz−2 dz = 0 shows that just because the function is
not analytic in the interior, the theorem need not failautomatically. This is quite common in mathematics and in life.If your hypotheses are satisfied, then you can say something withconfidence. But if not, it’s often “anything goes”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 227: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/227.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Domains With and Without Holes1. The Cauchy-Goursat Theorem works as long as the function is
analytic on a domain that contains the contour and the contour’sinterior.
2. But if C is the positively oriented unit circle, then∫C
z−1 dz = 2πi 6= 0 shows that the result need not hold when the
function is not analytic in the whole interior.
3. Side note:∫
Cz−2 dz = 0 shows that just because the function is
not analytic in the interior, the theorem need not failautomatically. This is quite common in mathematics and in life.If your hypotheses are satisfied, then you can say something withconfidence. But if not, it’s often “anything goes”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 228: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/228.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Domains With and Without Holes1. The Cauchy-Goursat Theorem works as long as the function is
analytic on a domain that contains the contour and the contour’sinterior.
2. But if C is the positively oriented unit circle, then∫C
z−1 dz = 2πi 6= 0 shows that the result need not hold when the
function is not analytic in the whole interior.
3. Side note:∫
Cz−2 dz = 0 shows that just because the function is
not analytic in the interior, the theorem need not failautomatically. This is quite common in mathematics and in life.If your hypotheses are satisfied, then you can say something withconfidence. But if not, it’s often “anything goes”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 229: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/229.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Domains With and Without Holes1. The Cauchy-Goursat Theorem works as long as the function is
analytic on a domain that contains the contour and the contour’sinterior.
2. But if C is the positively oriented unit circle, then∫C
z−1 dz = 2πi 6= 0 shows that the result need not hold when the
function is not analytic in the whole interior.
3. Side note:∫
Cz−2 dz = 0 shows that just because the function is
not analytic in the interior, the theorem need not failautomatically.
This is quite common in mathematics and in life.If your hypotheses are satisfied, then you can say something withconfidence. But if not, it’s often “anything goes”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 230: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/230.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Domains With and Without Holes1. The Cauchy-Goursat Theorem works as long as the function is
analytic on a domain that contains the contour and the contour’sinterior.
2. But if C is the positively oriented unit circle, then∫C
z−1 dz = 2πi 6= 0 shows that the result need not hold when the
function is not analytic in the whole interior.
3. Side note:∫
Cz−2 dz = 0 shows that just because the function is
not analytic in the interior, the theorem need not failautomatically. This is quite common in mathematics and in life.
If your hypotheses are satisfied, then you can say something withconfidence. But if not, it’s often “anything goes”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 231: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/231.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Domains With and Without Holes1. The Cauchy-Goursat Theorem works as long as the function is
analytic on a domain that contains the contour and the contour’sinterior.
2. But if C is the positively oriented unit circle, then∫C
z−1 dz = 2πi 6= 0 shows that the result need not hold when the
function is not analytic in the whole interior.
3. Side note:∫
Cz−2 dz = 0 shows that just because the function is
not analytic in the interior, the theorem need not failautomatically. This is quite common in mathematics and in life.If your hypotheses are satisfied, then you can say something withconfidence.
But if not, it’s often “anything goes”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 232: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/232.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Domains With and Without Holes1. The Cauchy-Goursat Theorem works as long as the function is
analytic on a domain that contains the contour and the contour’sinterior.
2. But if C is the positively oriented unit circle, then∫C
z−1 dz = 2πi 6= 0 shows that the result need not hold when the
function is not analytic in the whole interior.
3. Side note:∫
Cz−2 dz = 0 shows that just because the function is
not analytic in the interior, the theorem need not failautomatically. This is quite common in mathematics and in life.If your hypotheses are satisfied, then you can say something withconfidence. But if not, it’s often “anything goes”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 233: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/233.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Domains With and Without Holes
4. The problem with z−1 on the unit circle is apparently that thefunction is not analytic (not even defined) at z = 0.
5. Pictorially, the domain of z−1 has a “hole” at zero, and we wantto formalize that idea.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 234: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/234.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Domains With and Without Holes4. The problem with z−1 on the unit circle is apparently that the
function is not analytic (not even defined) at z = 0.
5. Pictorially, the domain of z−1 has a “hole” at zero, and we wantto formalize that idea.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 235: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/235.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Domains With and Without Holes4. The problem with z−1 on the unit circle is apparently that the
function is not analytic (not even defined) at z = 0.5. Pictorially, the domain of z−1 has a “hole” at zero
, and we wantto formalize that idea.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 236: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/236.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Domains With and Without Holes4. The problem with z−1 on the unit circle is apparently that the
function is not analytic (not even defined) at z = 0.5. Pictorially, the domain of z−1 has a “hole” at zero, and we want
to formalize that idea.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 237: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/237.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Definition.
A domain D (remember domains are connected) so thatfor every simple closed contour C in D the interior of C is containedin D, too, is called simply connected.
D
? �C
Simply connected means “no holes”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 238: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/238.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Definition. A domain D
(remember domains are connected) so thatfor every simple closed contour C in D the interior of C is containedin D, too, is called simply connected.
D
? �C
Simply connected means “no holes”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 239: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/239.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Definition. A domain D (remember domains are connected)
so thatfor every simple closed contour C in D the interior of C is containedin D, too, is called simply connected.
D
? �C
Simply connected means “no holes”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 240: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/240.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Definition. A domain D (remember domains are connected) so thatfor every simple closed contour C in D the interior of C is containedin D, too, is called simply connected.
D
? �C
Simply connected means “no holes”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 241: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/241.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Definition. A domain D (remember domains are connected) so thatfor every simple closed contour C in D the interior of C is containedin D, too, is called simply connected.
D
? �C
Simply connected means “no holes”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 242: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/242.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Definition. A domain D (remember domains are connected) so thatfor every simple closed contour C in D the interior of C is containedin D, too, is called simply connected.
D
? �
C
Simply connected means “no holes”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 243: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/243.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Definition. A domain D (remember domains are connected) so thatfor every simple closed contour C in D the interior of C is containedin D, too, is called simply connected.
D
? �C
Simply connected means “no holes”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 244: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/244.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Definition. A domain D (remember domains are connected) so thatfor every simple closed contour C in D the interior of C is containedin D, too, is called simply connected.
D
? �C
Simply connected means “no holes”.Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 245: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/245.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem.
If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 246: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/246.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 247: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/247.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof.
For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 248: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/248.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof. For simple closed contours, this is the Cauchy-GoursatTheorem.
Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 249: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/249.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 250: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/250.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 251: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/251.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 252: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/252.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 253: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/253.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 254: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/254.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 255: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/255.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 256: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/256.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>
U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 257: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/257.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 258: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/258.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 259: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/259.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 260: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/260.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 261: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/261.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then for
any closed contour C in D we have∫
Cf (z) dz = 0.
Proof. For simple closed contours, this is the Cauchy-GoursatTheorem. Contours that intersect themselves can be broken up intosimple contours.
-
6ℑ(z)
ℜ(z)
1
?
}
>
U
y
>U
)
Y
C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 262: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/262.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem.
If f is analytic on the simply connected domain D, then fhas an antiderivative.
Proof. By the preceding theorem, integrals of f over closed contoursare zero. By our first theorem, f must have an antiderivative.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 263: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/263.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then fhas an antiderivative.
Proof. By the preceding theorem, integrals of f over closed contoursare zero. By our first theorem, f must have an antiderivative.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 264: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/264.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then fhas an antiderivative.
Proof.
By the preceding theorem, integrals of f over closed contoursare zero. By our first theorem, f must have an antiderivative.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 265: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/265.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then fhas an antiderivative.
Proof. By the preceding theorem, integrals of f over closed contoursare zero.
By our first theorem, f must have an antiderivative.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 266: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/266.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then fhas an antiderivative.
Proof. By the preceding theorem, integrals of f over closed contoursare zero. By our first theorem, f must have an antiderivative.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 267: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/267.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. If f is analytic on the simply connected domain D, then fhas an antiderivative.
Proof. By the preceding theorem, integrals of f over closed contoursare zero. By our first theorem, f must have an antiderivative.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 268: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/268.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Definition.
A domain D the is not simply connected is calledmultiply connected.
D
hole? �
C
Multiply connected means there are holes.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 269: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/269.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Definition. A domain D the is not simply connected is calledmultiply connected.
D
hole? �
C
Multiply connected means there are holes.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 270: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/270.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Definition. A domain D the is not simply connected is calledmultiply connected.
D
hole? �
C
Multiply connected means there are holes.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 271: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/271.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Definition. A domain D the is not simply connected is calledmultiply connected.
D
hole? �
C
Multiply connected means there are holes.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 272: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/272.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Definition. A domain D the is not simply connected is calledmultiply connected.
D
hole
? �C
Multiply connected means there are holes.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 273: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/273.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Definition. A domain D the is not simply connected is calledmultiply connected.
D
hole? �
C
Multiply connected means there are holes.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 274: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/274.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Definition. A domain D the is not simply connected is calledmultiply connected.
D
hole? �
C
Multiply connected means there are holes.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 275: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/275.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Definition. A domain D the is not simply connected is calledmultiply connected.
D
hole? �
C
Multiply connected means there are holes.Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 276: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/276.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem.
Let C be a positively oriented simple closed contour. LetC1, . . . ,Cn be pairwise disjoint clockwise (that is, negatively) orientedsimple closed contours in the interior of C. Let f be analytic in a(possibly multiply connected) domain that contains the contours andthe region inside C and outside the Cj. Then∫
Cf (z) dz+
n
∑j=1
∫Cj
f (z) dz = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 277: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/277.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C be a positively oriented simple closed contour.
LetC1, . . . ,Cn be pairwise disjoint clockwise (that is, negatively) orientedsimple closed contours in the interior of C. Let f be analytic in a(possibly multiply connected) domain that contains the contours andthe region inside C and outside the Cj. Then∫
Cf (z) dz+
n
∑j=1
∫Cj
f (z) dz = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 278: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/278.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C be a positively oriented simple closed contour. LetC1, . . . ,Cn be pairwise disjoint clockwise (that is, negatively) orientedsimple closed contours in the interior of C.
Let f be analytic in a(possibly multiply connected) domain that contains the contours andthe region inside C and outside the Cj. Then∫
Cf (z) dz+
n
∑j=1
∫Cj
f (z) dz = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 279: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/279.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C be a positively oriented simple closed contour. LetC1, . . . ,Cn be pairwise disjoint clockwise (that is, negatively) orientedsimple closed contours in the interior of C. Let f be analytic in a(possibly multiply connected) domain that contains the contours andthe region inside C and outside the Cj.
Then∫C
f (z) dz+n
∑j=1
∫Cj
f (z) dz = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 280: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/280.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C be a positively oriented simple closed contour. LetC1, . . . ,Cn be pairwise disjoint clockwise (that is, negatively) orientedsimple closed contours in the interior of C. Let f be analytic in a(possibly multiply connected) domain that contains the contours andthe region inside C and outside the Cj. Then∫
Cf (z) dz+
n
∑j=1
∫Cj
f (z) dz = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 281: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/281.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
D
�
�
�
-
-
1
]
�
C
-
�
?O
rC1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 282: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/282.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
D
�
�
�
-
-
1
]
�
C
-
�
?O
rC1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 283: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/283.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 284: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/284.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
D
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1
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C
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 285: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/285.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
D
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-
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1
]
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C
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?O
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 286: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/286.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
D
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�
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-
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1
]
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C
-
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?O
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C1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 287: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/287.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
D
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�
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-
-
1
]
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C
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?O
rC1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 288: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/288.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
D
r-�
?O
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-
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1
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 289: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/289.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
D
r-�
?O
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-
-
1
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C
C1
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 290: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/290.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
D
r-�
?O
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-
-
1
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C
C1
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 291: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/291.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
D
r-�
?O
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-
-
1
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C
C1
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 292: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/292.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
D
r-�
?O
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-
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1
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C1
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 293: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/293.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
D
r-�
?O
�
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-
-
1
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C
C1
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 294: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/294.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
D
r-�
?O
�
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-
-
1
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C
C1
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 295: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/295.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Proof.
D
r-�
?O
�
�
�
-
-
1
]
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C
C1
rrrrIR
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 296: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/296.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem.
Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between
them. Then∫
C1
f (z) dz =∫
C2
f (z) dz.
Proof. Same proof as previous result, except that, because bothcontours are positively oriented, this time the difference is zero. Nowbring the integral over C2 to the right side.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 297: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/297.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2.
Let f beanalytic in a region that contains the contours and the region between
them. Then∫
C1
f (z) dz =∫
C2
f (z) dz.
Proof. Same proof as previous result, except that, because bothcontours are positively oriented, this time the difference is zero. Nowbring the integral over C2 to the right side.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 298: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/298.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between
them.
Then∫
C1
f (z) dz =∫
C2
f (z) dz.
Proof. Same proof as previous result, except that, because bothcontours are positively oriented, this time the difference is zero. Nowbring the integral over C2 to the right side.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 299: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/299.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between
them. Then∫
C1
f (z) dz =∫
C2
f (z) dz.
Proof. Same proof as previous result, except that, because bothcontours are positively oriented, this time the difference is zero. Nowbring the integral over C2 to the right side.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 300: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/300.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between
them. Then∫
C1
f (z) dz =∫
C2
f (z) dz.
Proof.
Same proof as previous result, except that, because bothcontours are positively oriented, this time the difference is zero. Nowbring the integral over C2 to the right side.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 301: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/301.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between
them. Then∫
C1
f (z) dz =∫
C2
f (z) dz.
Proof. Same proof as previous result
, except that, because bothcontours are positively oriented, this time the difference is zero. Nowbring the integral over C2 to the right side.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 302: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/302.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between
them. Then∫
C1
f (z) dz =∫
C2
f (z) dz.
Proof. Same proof as previous result, except that, because bothcontours are positively oriented
, this time the difference is zero. Nowbring the integral over C2 to the right side.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 303: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/303.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between
them. Then∫
C1
f (z) dz =∫
C2
f (z) dz.
Proof. Same proof as previous result, except that, because bothcontours are positively oriented, this time the difference is zero.
Nowbring the integral over C2 to the right side.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 304: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/304.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between
them. Then∫
C1
f (z) dz =∫
C2
f (z) dz.
Proof. Same proof as previous result, except that, because bothcontours are positively oriented, this time the difference is zero. Nowbring the integral over C2 to the right side.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 305: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/305.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Theorem. Let C1 and C2 be positively oriented simple closedcontours so that C1 is contained in the interior of C2. Let f beanalytic in a region that contains the contours and the region between
them. Then∫
C1
f (z) dz =∫
C2
f (z) dz.
Proof. Same proof as previous result, except that, because bothcontours are positively oriented, this time the difference is zero. Nowbring the integral over C2 to the right side.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 306: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/306.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example.
The integral of z−1 around any positively oriented simpleclosed contour that has the origin in its interior is 2πi.
We have proved that the integral of this function along positivelyoriented circles around the origin is 2πi. The preceding theorem letsus go to arbitrary positively oriented simple closed contours that havethe origin in their interior, because we can always put a tiny circle intothe contour’s interior or draw a large circle around it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 307: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/307.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of z−1 around any positively oriented simpleclosed contour that has the origin in its interior is 2πi.
We have proved that the integral of this function along positivelyoriented circles around the origin is 2πi. The preceding theorem letsus go to arbitrary positively oriented simple closed contours that havethe origin in their interior, because we can always put a tiny circle intothe contour’s interior or draw a large circle around it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 308: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/308.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of z−1 around any positively oriented simpleclosed contour that has the origin in its interior is 2πi.
We have proved that the integral of this function along positivelyoriented circles around the origin is 2πi.
The preceding theorem letsus go to arbitrary positively oriented simple closed contours that havethe origin in their interior, because we can always put a tiny circle intothe contour’s interior or draw a large circle around it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 309: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/309.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of z−1 around any positively oriented simpleclosed contour that has the origin in its interior is 2πi.
We have proved that the integral of this function along positivelyoriented circles around the origin is 2πi. The preceding theorem letsus go to arbitrary positively oriented simple closed contours that havethe origin in their interior
, because we can always put a tiny circle intothe contour’s interior or draw a large circle around it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems
![Page 310: Integral Theorems · 2010. 9. 15. · logo1 AntiderivativesCauchy-Goursat TheoremTwo Kinds of Domains Introduction 1.Much of the strength of complex analysis derives from the fact](https://reader035.vdocuments.us/reader035/viewer/2022063016/5fd4e69cfbc1e17c597a628c/html5/thumbnails/310.jpg)
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Example. The integral of z−1 around any positively oriented simpleclosed contour that has the origin in its interior is 2πi.
We have proved that the integral of this function along positivelyoriented circles around the origin is 2πi. The preceding theorem letsus go to arbitrary positively oriented simple closed contours that havethe origin in their interior, because we can always put a tiny circle intothe contour’s interior or draw a large circle around it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Integral Theorems