the residue theorem · 2010-10-04 · 5.we will prove the requisite theorem (the residue theorem)...
TRANSCRIPT
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
The Residue Theorem
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Introduction
1. The Cauchy-Goursat Theorem says that if a function is analyticon and in a closed contour C, then the integral over the closedcontour is zero.
2. But what if the function is not analytic?3. We will avoid situations where the function “blows up” (goes to
infinity) on the contour. So we will not need to generalizecontour integrals to “improper contour integrals”.
4. But the situation in which the function is not analytic inside thecontour turns out to be quite interesting.
5. We will prove the requisite theorem (the Residue Theorem) inthis presentation and we will also lay the abstract groundwork.
6. We will then spend an extensive amount of time with examplesthat show how widely applicable the Residue Theorem is.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Introduction1. The Cauchy-Goursat Theorem says that if a function is analytic
on and in a closed contour C, then the integral over the closedcontour is zero.
2. But what if the function is not analytic?3. We will avoid situations where the function “blows up” (goes to
infinity) on the contour. So we will not need to generalizecontour integrals to “improper contour integrals”.
4. But the situation in which the function is not analytic inside thecontour turns out to be quite interesting.
5. We will prove the requisite theorem (the Residue Theorem) inthis presentation and we will also lay the abstract groundwork.
6. We will then spend an extensive amount of time with examplesthat show how widely applicable the Residue Theorem is.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Introduction1. The Cauchy-Goursat Theorem says that if a function is analytic
on and in a closed contour C, then the integral over the closedcontour is zero.
2. But what if the function is not analytic?
3. We will avoid situations where the function “blows up” (goes toinfinity) on the contour. So we will not need to generalizecontour integrals to “improper contour integrals”.
4. But the situation in which the function is not analytic inside thecontour turns out to be quite interesting.
5. We will prove the requisite theorem (the Residue Theorem) inthis presentation and we will also lay the abstract groundwork.
6. We will then spend an extensive amount of time with examplesthat show how widely applicable the Residue Theorem is.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Introduction1. The Cauchy-Goursat Theorem says that if a function is analytic
on and in a closed contour C, then the integral over the closedcontour is zero.
2. But what if the function is not analytic?3. We will avoid situations where the function “blows up” (goes to
infinity) on the contour.
So we will not need to generalizecontour integrals to “improper contour integrals”.
4. But the situation in which the function is not analytic inside thecontour turns out to be quite interesting.
5. We will prove the requisite theorem (the Residue Theorem) inthis presentation and we will also lay the abstract groundwork.
6. We will then spend an extensive amount of time with examplesthat show how widely applicable the Residue Theorem is.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Introduction1. The Cauchy-Goursat Theorem says that if a function is analytic
on and in a closed contour C, then the integral over the closedcontour is zero.
2. But what if the function is not analytic?3. We will avoid situations where the function “blows up” (goes to
infinity) on the contour. So we will not need to generalizecontour integrals to “improper contour integrals”.
4. But the situation in which the function is not analytic inside thecontour turns out to be quite interesting.
5. We will prove the requisite theorem (the Residue Theorem) inthis presentation and we will also lay the abstract groundwork.
6. We will then spend an extensive amount of time with examplesthat show how widely applicable the Residue Theorem is.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Introduction1. The Cauchy-Goursat Theorem says that if a function is analytic
on and in a closed contour C, then the integral over the closedcontour is zero.
2. But what if the function is not analytic?3. We will avoid situations where the function “blows up” (goes to
infinity) on the contour. So we will not need to generalizecontour integrals to “improper contour integrals”.
4. But the situation in which the function is not analytic inside thecontour turns out to be quite interesting.
5. We will prove the requisite theorem (the Residue Theorem) inthis presentation and we will also lay the abstract groundwork.
6. We will then spend an extensive amount of time with examplesthat show how widely applicable the Residue Theorem is.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Introduction1. The Cauchy-Goursat Theorem says that if a function is analytic
on and in a closed contour C, then the integral over the closedcontour is zero.
2. But what if the function is not analytic?3. We will avoid situations where the function “blows up” (goes to
infinity) on the contour. So we will not need to generalizecontour integrals to “improper contour integrals”.
4. But the situation in which the function is not analytic inside thecontour turns out to be quite interesting.
5. We will prove the requisite theorem
(the Residue Theorem) inthis presentation and we will also lay the abstract groundwork.
6. We will then spend an extensive amount of time with examplesthat show how widely applicable the Residue Theorem is.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Introduction1. The Cauchy-Goursat Theorem says that if a function is analytic
on and in a closed contour C, then the integral over the closedcontour is zero.
2. But what if the function is not analytic?3. We will avoid situations where the function “blows up” (goes to
infinity) on the contour. So we will not need to generalizecontour integrals to “improper contour integrals”.
4. But the situation in which the function is not analytic inside thecontour turns out to be quite interesting.
5. We will prove the requisite theorem (the Residue Theorem)
inthis presentation and we will also lay the abstract groundwork.
6. We will then spend an extensive amount of time with examplesthat show how widely applicable the Residue Theorem is.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Introduction1. The Cauchy-Goursat Theorem says that if a function is analytic
on and in a closed contour C, then the integral over the closedcontour is zero.
2. But what if the function is not analytic?3. We will avoid situations where the function “blows up” (goes to
infinity) on the contour. So we will not need to generalizecontour integrals to “improper contour integrals”.
4. But the situation in which the function is not analytic inside thecontour turns out to be quite interesting.
5. We will prove the requisite theorem (the Residue Theorem) inthis presentation and we will also lay the abstract groundwork.
6. We will then spend an extensive amount of time with examplesthat show how widely applicable the Residue Theorem is.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Introduction1. The Cauchy-Goursat Theorem says that if a function is analytic
on and in a closed contour C, then the integral over the closedcontour is zero.
2. But what if the function is not analytic?3. We will avoid situations where the function “blows up” (goes to
infinity) on the contour. So we will not need to generalizecontour integrals to “improper contour integrals”.
4. But the situation in which the function is not analytic inside thecontour turns out to be quite interesting.
5. We will prove the requisite theorem (the Residue Theorem) inthis presentation and we will also lay the abstract groundwork.
6. We will then spend an extensive amount of time with examplesthat show how widely applicable the Residue Theorem is.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example.
The function f (z) =1
(z−1)(1+ z2)is not analytic at
z = 1, i,−i.
-
6ℑ(z)
ℜ(z)1−1
i
−i
rr
r
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
(z−1)(1+ z2)is not analytic at
z = 1, i,−i.
-
6ℑ(z)
ℜ(z)1−1
i
−i
rr
r
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
(z−1)(1+ z2)is not analytic at
z = 1, i,−i.
-
6ℑ(z)
ℜ(z)
1−1
i
−i
rr
r
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
(z−1)(1+ z2)is not analytic at
z = 1, i,−i.
-
6ℑ(z)
ℜ(z)
1−1
i
−i
rr
r
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
(z−1)(1+ z2)is not analytic at
z = 1, i,−i.
-
6ℑ(z)
ℜ(z)
1−1
i
−i
rr
r
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
(z−1)(1+ z2)is not analytic at
z = 1, i,−i.
-
6ℑ(z)
ℜ(z)
1−1
i
−i
rr
r
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
(z−1)(1+ z2)is not analytic at
z = 1, i,−i.
-
6ℑ(z)
ℜ(z)
1−1
i
−i
rr
r
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
(z−1)(1+ z2)is not analytic at
z = 1, i,−i.
-
6ℑ(z)
ℜ(z)1
−1
i
−i
rr
r
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
(z−1)(1+ z2)is not analytic at
z = 1, i,−i.
-
6ℑ(z)
ℜ(z)1−1
i
−i
rr
r
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
(z−1)(1+ z2)is not analytic at
z = 1, i,−i.
-
6ℑ(z)
ℜ(z)1−1
i
−i
rr
r
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
(z−1)(1+ z2)is not analytic at
z = 1, i,−i.
-
6ℑ(z)
ℜ(z)1−1
i
−i
rr
r
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
(z−1)(1+ z2)is not analytic at
z = 1, i,−i.
-
6ℑ(z)
ℜ(z)1−1
i
−i
r
r
r
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
(z−1)(1+ z2)is not analytic at
z = 1, i,−i.
-
6ℑ(z)
ℜ(z)1−1
i
−i
rr
r
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
(z−1)(1+ z2)is not analytic at
z = 1, i,−i.
-
6ℑ(z)
ℜ(z)1−1
i
−i
rr
r
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example.
The function f (z) =1
sin(
π
z
) is not analytic at
z = 1,12,13,14, . . . and at 0 and at z =−1,−1
2,−1
3,−1
4, . . ..
-
6ℑ(z)
ℜ(z)1−1
i
−i
rrr· · ·rr r r· · ·
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
sin(
π
z
) is not analytic at
z = 1,12,13,14, . . . and at 0
and at z =−1,−12,−1
3,−1
4, . . ..
-
6ℑ(z)
ℜ(z)1−1
i
−i
rrr· · ·rr r r· · ·
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
sin(
π
z
) is not analytic at
z = 1,12,13,14, . . . and at 0 and at z =−1,−1
2,−1
3,−1
4, . . ..
-
6ℑ(z)
ℜ(z)1−1
i
−i
rrr· · ·rr r r· · ·
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
sin(
π
z
) is not analytic at
z = 1,12,13,14, . . . and at 0 and at z =−1,−1
2,−1
3,−1
4, . . ..
-
6ℑ(z)
ℜ(z)1−1
i
−i
rrr· · ·rr r r· · ·
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
sin(
π
z
) is not analytic at
z = 1,12,13,14, . . . and at 0 and at z =−1,−1
2,−1
3,−1
4, . . ..
-
6ℑ(z)
ℜ(z)1−1
i
−i
r
rr· · ·rr r r· · ·
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
sin(
π
z
) is not analytic at
z = 1,12,13,14, . . . and at 0 and at z =−1,−1
2,−1
3,−1
4, . . ..
-
6ℑ(z)
ℜ(z)1−1
i
−i
rr
r· · ·rr r r· · ·
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
sin(
π
z
) is not analytic at
z = 1,12,13,14, . . . and at 0 and at z =−1,−1
2,−1
3,−1
4, . . ..
-
6ℑ(z)
ℜ(z)1−1
i
−i
rrr
· · ·rr r r· · ·
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
sin(
π
z
) is not analytic at
z = 1,12,13,14, . . . and at 0 and at z =−1,−1
2,−1
3,−1
4, . . ..
-
6ℑ(z)
ℜ(z)1−1
i
−i
rrr· · ·
rr r r· · ·
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
sin(
π
z
) is not analytic at
z = 1,12,13,14, . . . and at 0 and at z =−1,−1
2,−1
3,−1
4, . . ..
-
6ℑ(z)
ℜ(z)1−1
i
−i
rrr· · ·r
r r r· · ·
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
sin(
π
z
) is not analytic at
z = 1,12,13,14, . . . and at 0 and at z =−1,−1
2,−1
3,−1
4, . . ..
-
6ℑ(z)
ℜ(z)1−1
i
−i
rrr· · ·rr
r r· · ·
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
sin(
π
z
) is not analytic at
z = 1,12,13,14, . . . and at 0 and at z =−1,−1
2,−1
3,−1
4, . . ..
-
6ℑ(z)
ℜ(z)1−1
i
−i
rrr· · ·rr r
r· · ·
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
sin(
π
z
) is not analytic at
z = 1,12,13,14, . . . and at 0 and at z =−1,−1
2,−1
3,−1
4, . . ..
-
6ℑ(z)
ℜ(z)1−1
i
−i
rrr· · ·rr r r
· · ·
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
sin(
π
z
) is not analytic at
z = 1,12,13,14, . . . and at 0 and at z =−1,−1
2,−1
3,−1
4, . . ..
-
6ℑ(z)
ℜ(z)1−1
i
−i
rrr· · ·rr r r· · ·
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition.
If the function f is analytic for 0 < |z− z0|< R, then z0 iscalled an isolated singularity of f .
Notes.1. z = 1, i,−i are isolated singularities of f (z) =
1(z−1)(1+ z2)
.
2. z = 1,12,13,14, . . . are isolated singularities of f (z) =
1sin(
π
z
) .
But 0 is not an isolated singularity of f (z) =1
sin(
π
z
) .
3. 0 is not an isolated singularity of f (z) = Log(z) or of any rootfunction. (Remember that every branch cut must contain zero, sothese functions will not be analytic on a set 0 < |z|< R.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. If the function f is analytic for 0 < |z− z0|< R
, then z0 iscalled an isolated singularity of f .
Notes.1. z = 1, i,−i are isolated singularities of f (z) =
1(z−1)(1+ z2)
.
2. z = 1,12,13,14, . . . are isolated singularities of f (z) =
1sin(
π
z
) .
But 0 is not an isolated singularity of f (z) =1
sin(
π
z
) .
3. 0 is not an isolated singularity of f (z) = Log(z) or of any rootfunction. (Remember that every branch cut must contain zero, sothese functions will not be analytic on a set 0 < |z|< R.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. If the function f is analytic for 0 < |z− z0|< R, then z0 iscalled an isolated singularity of f .
Notes.1. z = 1, i,−i are isolated singularities of f (z) =
1(z−1)(1+ z2)
.
2. z = 1,12,13,14, . . . are isolated singularities of f (z) =
1sin(
π
z
) .
But 0 is not an isolated singularity of f (z) =1
sin(
π
z
) .
3. 0 is not an isolated singularity of f (z) = Log(z) or of any rootfunction. (Remember that every branch cut must contain zero, sothese functions will not be analytic on a set 0 < |z|< R.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. If the function f is analytic for 0 < |z− z0|< R, then z0 iscalled an isolated singularity of f .
Notes.
1. z = 1, i,−i are isolated singularities of f (z) =1
(z−1)(1+ z2).
2. z = 1,12,13,14, . . . are isolated singularities of f (z) =
1sin(
π
z
) .
But 0 is not an isolated singularity of f (z) =1
sin(
π
z
) .
3. 0 is not an isolated singularity of f (z) = Log(z) or of any rootfunction. (Remember that every branch cut must contain zero, sothese functions will not be analytic on a set 0 < |z|< R.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. If the function f is analytic for 0 < |z− z0|< R, then z0 iscalled an isolated singularity of f .
Notes.1. z = 1, i,−i are isolated singularities of f (z) =
1(z−1)(1+ z2)
.
2. z = 1,12,13,14, . . . are isolated singularities of f (z) =
1sin(
π
z
) .
But 0 is not an isolated singularity of f (z) =1
sin(
π
z
) .
3. 0 is not an isolated singularity of f (z) = Log(z) or of any rootfunction. (Remember that every branch cut must contain zero, sothese functions will not be analytic on a set 0 < |z|< R.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. If the function f is analytic for 0 < |z− z0|< R, then z0 iscalled an isolated singularity of f .
Notes.1. z = 1, i,−i are isolated singularities of f (z) =
1(z−1)(1+ z2)
.
2. z = 1,12,13,14, . . . are isolated singularities of f (z) =
1sin(
π
z
) .
But 0 is not an isolated singularity of f (z) =1
sin(
π
z
) .
3. 0 is not an isolated singularity of f (z) = Log(z) or of any rootfunction. (Remember that every branch cut must contain zero, sothese functions will not be analytic on a set 0 < |z|< R.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. If the function f is analytic for 0 < |z− z0|< R, then z0 iscalled an isolated singularity of f .
Notes.1. z = 1, i,−i are isolated singularities of f (z) =
1(z−1)(1+ z2)
.
2. z = 1,12,13,14, . . . are isolated singularities of f (z) =
1sin(
π
z
) .
But 0 is not an isolated singularity of f (z) =1
sin(
π
z
) .
3. 0 is not an isolated singularity of f (z) = Log(z) or of any rootfunction. (Remember that every branch cut must contain zero, sothese functions will not be analytic on a set 0 < |z|< R.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. If the function f is analytic for 0 < |z− z0|< R, then z0 iscalled an isolated singularity of f .
Notes.1. z = 1, i,−i are isolated singularities of f (z) =
1(z−1)(1+ z2)
.
2. z = 1,12,13,14, . . . are isolated singularities of f (z) =
1sin(
π
z
) .
But 0 is not an isolated singularity of f (z) =1
sin(
π
z
) .
3. 0 is not an isolated singularity of f (z) = Log(z) or of any rootfunction.
(Remember that every branch cut must contain zero, sothese functions will not be analytic on a set 0 < |z|< R.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. If the function f is analytic for 0 < |z− z0|< R, then z0 iscalled an isolated singularity of f .
Notes.1. z = 1, i,−i are isolated singularities of f (z) =
1(z−1)(1+ z2)
.
2. z = 1,12,13,14, . . . are isolated singularities of f (z) =
1sin(
π
z
) .
But 0 is not an isolated singularity of f (z) =1
sin(
π
z
) .
3. 0 is not an isolated singularity of f (z) = Log(z) or of any rootfunction. (Remember that every branch cut must contain zero
, sothese functions will not be analytic on a set 0 < |z|< R.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. If the function f is analytic for 0 < |z− z0|< R, then z0 iscalled an isolated singularity of f .
Notes.1. z = 1, i,−i are isolated singularities of f (z) =
1(z−1)(1+ z2)
.
2. z = 1,12,13,14, . . . are isolated singularities of f (z) =
1sin(
π
z
) .
But 0 is not an isolated singularity of f (z) =1
sin(
π
z
) .
3. 0 is not an isolated singularity of f (z) = Log(z) or of any rootfunction. (Remember that every branch cut must contain zero, sothese functions will not be analytic on a set 0 < |z|< R.)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition.
If the function f is analytic for 0 < |z− z0|< R, then it has
a Laurent expansion∞
∑n=−∞
cn(z− z0)n about z0. The coefficient c−1 is
called the residue of f at z0. It is also denoted Resz=z0(f ) := c−1.
The next result will show the relevance of residues.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. If the function f is analytic for 0 < |z− z0|< R
, then it has
a Laurent expansion∞
∑n=−∞
cn(z− z0)n about z0. The coefficient c−1 is
called the residue of f at z0. It is also denoted Resz=z0(f ) := c−1.
The next result will show the relevance of residues.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. If the function f is analytic for 0 < |z− z0|< R, then it has
a Laurent expansion∞
∑n=−∞
cn(z− z0)n about z0.
The coefficient c−1 is
called the residue of f at z0. It is also denoted Resz=z0(f ) := c−1.
The next result will show the relevance of residues.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. If the function f is analytic for 0 < |z− z0|< R, then it has
a Laurent expansion∞
∑n=−∞
cn(z− z0)n about z0. The coefficient c−1 is
called the residue of f at z0.
It is also denoted Resz=z0(f ) := c−1.
The next result will show the relevance of residues.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. If the function f is analytic for 0 < |z− z0|< R, then it has
a Laurent expansion∞
∑n=−∞
cn(z− z0)n about z0. The coefficient c−1 is
called the residue of f at z0. It is also denoted Resz=z0(f ) := c−1.
The next result will show the relevance of residues.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. If the function f is analytic for 0 < |z− z0|< R, then it has
a Laurent expansion∞
∑n=−∞
cn(z− z0)n about z0. The coefficient c−1 is
called the residue of f at z0. It is also denoted Resz=z0(f ) := c−1.
The next result will show the relevance of residues.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem.
If the function f is analytic for 0 < |z− z0|< R and C is apositively oriented simple closed contour around z0 that is contained
in 0 < |z− z0|< R, then∫
Cf (z) dz = 2πiResz=z0(f ).
Proof. From the theorem on Laurent expansions, we have that
Resz=z0(f ) = a−1 =1
2πi
∮C
f (ξ )(ξ − z0)(−1)+1 dξ =
12πi
∮C
f (ξ ) dξ ,
where C is any circle around z0 with radius < R. Replacement of thecircle with any contour around the origin requires an argument similarto the one that shows that we can use circles of any radius.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. If the function f is analytic for 0 < |z− z0|< R
and C is apositively oriented simple closed contour around z0 that is contained
in 0 < |z− z0|< R, then∫
Cf (z) dz = 2πiResz=z0(f ).
Proof. From the theorem on Laurent expansions, we have that
Resz=z0(f ) = a−1 =1
2πi
∮C
f (ξ )(ξ − z0)(−1)+1 dξ =
12πi
∮C
f (ξ ) dξ ,
where C is any circle around z0 with radius < R. Replacement of thecircle with any contour around the origin requires an argument similarto the one that shows that we can use circles of any radius.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. If the function f is analytic for 0 < |z− z0|< R and C is apositively oriented simple closed contour around z0 that is contained
in 0 < |z− z0|< R
, then∫
Cf (z) dz = 2πiResz=z0(f ).
Proof. From the theorem on Laurent expansions, we have that
Resz=z0(f ) = a−1 =1
2πi
∮C
f (ξ )(ξ − z0)(−1)+1 dξ =
12πi
∮C
f (ξ ) dξ ,
where C is any circle around z0 with radius < R. Replacement of thecircle with any contour around the origin requires an argument similarto the one that shows that we can use circles of any radius.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. If the function f is analytic for 0 < |z− z0|< R and C is apositively oriented simple closed contour around z0 that is contained
in 0 < |z− z0|< R, then∫
Cf (z) dz = 2πiResz=z0(f ).
Proof. From the theorem on Laurent expansions, we have that
Resz=z0(f ) = a−1 =1
2πi
∮C
f (ξ )(ξ − z0)(−1)+1 dξ =
12πi
∮C
f (ξ ) dξ ,
where C is any circle around z0 with radius < R. Replacement of thecircle with any contour around the origin requires an argument similarto the one that shows that we can use circles of any radius.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. If the function f is analytic for 0 < |z− z0|< R and C is apositively oriented simple closed contour around z0 that is contained
in 0 < |z− z0|< R, then∫
Cf (z) dz = 2πiResz=z0(f ).
Proof.
From the theorem on Laurent expansions, we have that
Resz=z0(f ) = a−1 =1
2πi
∮C
f (ξ )(ξ − z0)(−1)+1 dξ =
12πi
∮C
f (ξ ) dξ ,
where C is any circle around z0 with radius < R. Replacement of thecircle with any contour around the origin requires an argument similarto the one that shows that we can use circles of any radius.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. If the function f is analytic for 0 < |z− z0|< R and C is apositively oriented simple closed contour around z0 that is contained
in 0 < |z− z0|< R, then∫
Cf (z) dz = 2πiResz=z0(f ).
Proof. From the theorem on Laurent expansions, we have that
Resz=z0(f )
= a−1 =1
2πi
∮C
f (ξ )(ξ − z0)(−1)+1 dξ =
12πi
∮C
f (ξ ) dξ ,
where C is any circle around z0 with radius < R. Replacement of thecircle with any contour around the origin requires an argument similarto the one that shows that we can use circles of any radius.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. If the function f is analytic for 0 < |z− z0|< R and C is apositively oriented simple closed contour around z0 that is contained
in 0 < |z− z0|< R, then∫
Cf (z) dz = 2πiResz=z0(f ).
Proof. From the theorem on Laurent expansions, we have that
Resz=z0(f ) = a−1
=1
2πi
∮C
f (ξ )(ξ − z0)(−1)+1 dξ =
12πi
∮C
f (ξ ) dξ ,
where C is any circle around z0 with radius < R. Replacement of thecircle with any contour around the origin requires an argument similarto the one that shows that we can use circles of any radius.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. If the function f is analytic for 0 < |z− z0|< R and C is apositively oriented simple closed contour around z0 that is contained
in 0 < |z− z0|< R, then∫
Cf (z) dz = 2πiResz=z0(f ).
Proof. From the theorem on Laurent expansions, we have that
Resz=z0(f ) = a−1 =1
2πi
∮C
f (ξ )(ξ − z0)(−1)+1 dξ
=1
2πi
∮C
f (ξ ) dξ ,
where C is any circle around z0 with radius < R. Replacement of thecircle with any contour around the origin requires an argument similarto the one that shows that we can use circles of any radius.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. If the function f is analytic for 0 < |z− z0|< R and C is apositively oriented simple closed contour around z0 that is contained
in 0 < |z− z0|< R, then∫
Cf (z) dz = 2πiResz=z0(f ).
Proof. From the theorem on Laurent expansions, we have that
Resz=z0(f ) = a−1 =1
2πi
∮C
f (ξ )(ξ − z0)(−1)+1 dξ =
12πi
∮C
f (ξ ) dξ
,
where C is any circle around z0 with radius < R. Replacement of thecircle with any contour around the origin requires an argument similarto the one that shows that we can use circles of any radius.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. If the function f is analytic for 0 < |z− z0|< R and C is apositively oriented simple closed contour around z0 that is contained
in 0 < |z− z0|< R, then∫
Cf (z) dz = 2πiResz=z0(f ).
Proof. From the theorem on Laurent expansions, we have that
Resz=z0(f ) = a−1 =1
2πi
∮C
f (ξ )(ξ − z0)(−1)+1 dξ =
12πi
∮C
f (ξ ) dξ ,
where C is any circle around z0 with radius < R.
Replacement of thecircle with any contour around the origin requires an argument similarto the one that shows that we can use circles of any radius.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. If the function f is analytic for 0 < |z− z0|< R and C is apositively oriented simple closed contour around z0 that is contained
in 0 < |z− z0|< R, then∫
Cf (z) dz = 2πiResz=z0(f ).
Proof. From the theorem on Laurent expansions, we have that
Resz=z0(f ) = a−1 =1
2πi
∮C
f (ξ )(ξ − z0)(−1)+1 dξ =
12πi
∮C
f (ξ ) dξ ,
where C is any circle around z0 with radius < R. Replacement of thecircle with any contour around the origin requires an argument similarto the one that shows that we can use circles of any radius.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof (concl.)
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof (concl.)
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof (concl.)
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof (concl.)
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof (concl.)
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof (concl.)
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof (concl.)
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof (concl.)
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof (concl.)
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Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example.
Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez
zdz = 2πi
The functionez
zhas only the singularity at 0 inside the contour. The
residue ofez
z=
1z
+∞
∑n=0
zn
(n+1)!at z = 0 is 1. Now apply the
preceding theorem.Note that direct computation gives the same result, because the
integral of∞
∑n=0
zn
(n+1)!over any closed contour is 0 and
∫C
1z
dz =∫ 2π
0
1eiθ ieiθ dθ = 2πi.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation.
Then∫
C
ez
zdz = 2πi
The functionez
zhas only the singularity at 0 inside the contour. The
residue ofez
z=
1z
+∞
∑n=0
zn
(n+1)!at z = 0 is 1. Now apply the
preceding theorem.Note that direct computation gives the same result, because the
integral of∞
∑n=0
zn
(n+1)!over any closed contour is 0 and
∫C
1z
dz =∫ 2π
0
1eiθ ieiθ dθ = 2πi.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez
zdz = 2πi
The functionez
zhas only the singularity at 0 inside the contour. The
residue ofez
z=
1z
+∞
∑n=0
zn
(n+1)!at z = 0 is 1. Now apply the
preceding theorem.Note that direct computation gives the same result, because the
integral of∞
∑n=0
zn
(n+1)!over any closed contour is 0 and
∫C
1z
dz =∫ 2π
0
1eiθ ieiθ dθ = 2πi.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez
zdz = 2πi
The functionez
zhas only the singularity at 0 inside the contour.
The
residue ofez
z=
1z
+∞
∑n=0
zn
(n+1)!at z = 0 is 1. Now apply the
preceding theorem.Note that direct computation gives the same result, because the
integral of∞
∑n=0
zn
(n+1)!over any closed contour is 0 and
∫C
1z
dz =∫ 2π
0
1eiθ ieiθ dθ = 2πi.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez
zdz = 2πi
The functionez
zhas only the singularity at 0 inside the contour. The
residue ofez
z
=1z
+∞
∑n=0
zn
(n+1)!at z = 0 is 1. Now apply the
preceding theorem.Note that direct computation gives the same result, because the
integral of∞
∑n=0
zn
(n+1)!over any closed contour is 0 and
∫C
1z
dz =∫ 2π
0
1eiθ ieiθ dθ = 2πi.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez
zdz = 2πi
The functionez
zhas only the singularity at 0 inside the contour. The
residue ofez
z=
1z
+∞
∑n=0
zn
(n+1)!
at z = 0 is 1. Now apply the
preceding theorem.Note that direct computation gives the same result, because the
integral of∞
∑n=0
zn
(n+1)!over any closed contour is 0 and
∫C
1z
dz =∫ 2π
0
1eiθ ieiθ dθ = 2πi.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez
zdz = 2πi
The functionez
zhas only the singularity at 0 inside the contour. The
residue ofez
z=
1z
+∞
∑n=0
zn
(n+1)!at z = 0 is 1.
Now apply the
preceding theorem.Note that direct computation gives the same result, because the
integral of∞
∑n=0
zn
(n+1)!over any closed contour is 0 and
∫C
1z
dz =∫ 2π
0
1eiθ ieiθ dθ = 2πi.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez
zdz = 2πi
The functionez
zhas only the singularity at 0 inside the contour. The
residue ofez
z=
1z
+∞
∑n=0
zn
(n+1)!at z = 0 is 1. Now apply the
preceding theorem.
Note that direct computation gives the same result, because the
integral of∞
∑n=0
zn
(n+1)!over any closed contour is 0 and
∫C
1z
dz =∫ 2π
0
1eiθ ieiθ dθ = 2πi.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez
zdz = 2πi
The functionez
zhas only the singularity at 0 inside the contour. The
residue ofez
z=
1z
+∞
∑n=0
zn
(n+1)!at z = 0 is 1. Now apply the
preceding theorem.Note that direct computation gives the same result
, because the
integral of∞
∑n=0
zn
(n+1)!over any closed contour is 0 and
∫C
1z
dz =∫ 2π
0
1eiθ ieiθ dθ = 2πi.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez
zdz = 2πi
The functionez
zhas only the singularity at 0 inside the contour. The
residue ofez
z=
1z
+∞
∑n=0
zn
(n+1)!at z = 0 is 1. Now apply the
preceding theorem.Note that direct computation gives the same result, because the
integral of∞
∑n=0
zn
(n+1)!over any closed contour is 0
and
∫C
1z
dz =∫ 2π
0
1eiθ ieiθ dθ = 2πi.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez
zdz = 2πi
The functionez
zhas only the singularity at 0 inside the contour. The
residue ofez
z=
1z
+∞
∑n=0
zn
(n+1)!at z = 0 is 1. Now apply the
preceding theorem.Note that direct computation gives the same result, because the
integral of∞
∑n=0
zn
(n+1)!over any closed contour is 0 and
∫C
1z
dz
=∫ 2π
0
1eiθ ieiθ dθ = 2πi.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez
zdz = 2πi
The functionez
zhas only the singularity at 0 inside the contour. The
residue ofez
z=
1z
+∞
∑n=0
zn
(n+1)!at z = 0 is 1. Now apply the
preceding theorem.Note that direct computation gives the same result, because the
integral of∞
∑n=0
zn
(n+1)!over any closed contour is 0 and
∫C
1z
dz =∫ 2π
0
1eiθ ieiθ dθ
= 2πi.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez
zdz = 2πi
The functionez
zhas only the singularity at 0 inside the contour. The
residue ofez
z=
1z
+∞
∑n=0
zn
(n+1)!at z = 0 is 1. Now apply the
preceding theorem.Note that direct computation gives the same result, because the
integral of∞
∑n=0
zn
(n+1)!over any closed contour is 0 and
∫C
1z
dz =∫ 2π
0
1eiθ ieiθ dθ = 2πi.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example.
Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez2
z2 dz = 0
The functionez2
z2 has only the singularity at 0 inside the contour. The
residue ofez2
z2 =1z2 +
∞
∑n=0
z2n
(n+1)!at z = 0 is 0. Now apply the
preceding theorem.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation.
Then∫
C
ez2
z2 dz = 0
The functionez2
z2 has only the singularity at 0 inside the contour. The
residue ofez2
z2 =1z2 +
∞
∑n=0
z2n
(n+1)!at z = 0 is 0. Now apply the
preceding theorem.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez2
z2 dz = 0
The functionez2
z2 has only the singularity at 0 inside the contour. The
residue ofez2
z2 =1z2 +
∞
∑n=0
z2n
(n+1)!at z = 0 is 0. Now apply the
preceding theorem.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez2
z2 dz = 0
The functionez2
z2 has only the singularity at 0 inside the contour.
The
residue ofez2
z2 =1z2 +
∞
∑n=0
z2n
(n+1)!at z = 0 is 0. Now apply the
preceding theorem.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez2
z2 dz = 0
The functionez2
z2 has only the singularity at 0 inside the contour. The
residue ofez2
z2
=1z2 +
∞
∑n=0
z2n
(n+1)!at z = 0 is 0. Now apply the
preceding theorem.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez2
z2 dz = 0
The functionez2
z2 has only the singularity at 0 inside the contour. The
residue ofez2
z2 =1z2 +
∞
∑n=0
z2n
(n+1)!
at z = 0 is 0. Now apply the
preceding theorem.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez2
z2 dz = 0
The functionez2
z2 has only the singularity at 0 inside the contour. The
residue ofez2
z2 =1z2 +
∞
∑n=0
z2n
(n+1)!at z = 0 is 0.
Now apply the
preceding theorem.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Let C be the unit circle, traversed in the positive
orientation. Then∫
C
ez2
z2 dz = 0
The functionez2
z2 has only the singularity at 0 inside the contour. The
residue ofez2
z2 =1z2 +
∞
∑n=0
z2n
(n+1)!at z = 0 is 0. Now apply the
preceding theorem.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem.
Residue Theorem. Let C be a simple closed positivelyoriented contour, let z1, . . . ,zn be points in the interior of C, and letthe function f be analytic on C and in its interior, except possibly atthe zj. Then
12πi
∫C
f (ξ ) dξ =n
∑j=1
Resz=zj(f ).
Proof. Let Cj be positively oriented circle around zj so that no two ofC1, . . . ,Cn intersect and so that all are contained in the interior of C.Then by extension of Cauchy-Goursat theorem∫
Cf (z) dz =
n
∑j=1
∫Cj
f (z) dz = 2πin
∑j=1
Resz=zj(f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. Residue Theorem. Let C be a simple closed positivelyoriented contour, let z1, . . . ,zn be points in the interior of C, and letthe function f be analytic on C and in its interior, except possibly atthe zj. Then
12πi
∫C
f (ξ ) dξ =n
∑j=1
Resz=zj(f ).
Proof. Let Cj be positively oriented circle around zj so that no two ofC1, . . . ,Cn intersect and so that all are contained in the interior of C.Then by extension of Cauchy-Goursat theorem∫
Cf (z) dz =
n
∑j=1
∫Cj
f (z) dz = 2πin
∑j=1
Resz=zj(f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. Residue Theorem. Let C be a simple closed positivelyoriented contour, let z1, . . . ,zn be points in the interior of C, and letthe function f be analytic on C and in its interior, except possibly atthe zj. Then
12πi
∫C
f (ξ ) dξ =n
∑j=1
Resz=zj(f ).
Proof.
Let Cj be positively oriented circle around zj so that no two ofC1, . . . ,Cn intersect and so that all are contained in the interior of C.Then by extension of Cauchy-Goursat theorem∫
Cf (z) dz =
n
∑j=1
∫Cj
f (z) dz = 2πin
∑j=1
Resz=zj(f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. Residue Theorem. Let C be a simple closed positivelyoriented contour, let z1, . . . ,zn be points in the interior of C, and letthe function f be analytic on C and in its interior, except possibly atthe zj. Then
12πi
∫C
f (ξ ) dξ =n
∑j=1
Resz=zj(f ).
Proof. Let Cj be positively oriented circle around zj
so that no two ofC1, . . . ,Cn intersect and so that all are contained in the interior of C.Then by extension of Cauchy-Goursat theorem∫
Cf (z) dz =
n
∑j=1
∫Cj
f (z) dz = 2πin
∑j=1
Resz=zj(f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. Residue Theorem. Let C be a simple closed positivelyoriented contour, let z1, . . . ,zn be points in the interior of C, and letthe function f be analytic on C and in its interior, except possibly atthe zj. Then
12πi
∫C
f (ξ ) dξ =n
∑j=1
Resz=zj(f ).
Proof. Let Cj be positively oriented circle around zj so that no two ofC1, . . . ,Cn intersect and so that all are contained in the interior of C.
Then by extension of Cauchy-Goursat theorem∫C
f (z) dz =n
∑j=1
∫Cj
f (z) dz = 2πin
∑j=1
Resz=zj(f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. Residue Theorem. Let C be a simple closed positivelyoriented contour, let z1, . . . ,zn be points in the interior of C, and letthe function f be analytic on C and in its interior, except possibly atthe zj. Then
12πi
∫C
f (ξ ) dξ =n
∑j=1
Resz=zj(f ).
Proof. Let Cj be positively oriented circle around zj so that no two ofC1, . . . ,Cn intersect and so that all are contained in the interior of C.Then by extension of Cauchy-Goursat theorem∫
Cf (z) dz =
n
∑j=1
∫Cj
f (z) dz
= 2πin
∑j=1
Resz=zj(f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. Residue Theorem. Let C be a simple closed positivelyoriented contour, let z1, . . . ,zn be points in the interior of C, and letthe function f be analytic on C and in its interior, except possibly atthe zj. Then
12πi
∫C
f (ξ ) dξ =n
∑j=1
Resz=zj(f ).
Proof. Let Cj be positively oriented circle around zj so that no two ofC1, . . . ,Cn intersect and so that all are contained in the interior of C.Then by extension of Cauchy-Goursat theorem∫
Cf (z) dz =
n
∑j=1
∫Cj
f (z) dz = 2πin
∑j=1
Resz=zj(f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. Residue Theorem. Let C be a simple closed positivelyoriented contour, let z1, . . . ,zn be points in the interior of C, and letthe function f be analytic on C and in its interior, except possibly atthe zj. Then
12πi
∫C
f (ξ ) dξ =n
∑j=1
Resz=zj(f ).
Proof. Let Cj be positively oriented circle around zj so that no two ofC1, . . . ,Cn intersect and so that all are contained in the interior of C.Then by extension of Cauchy-Goursat theorem∫
Cf (z) dz =
n
∑j=1
∫Cj
f (z) dz = 2πin
∑j=1
Resz=zj(f ).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example.
Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)
6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
r
i
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r
−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z+ i
1z− i
Resz=i
(z2
z2 +1
)= − 1
i+ i=
i2
Residue at −i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z− i
1z+ i
Resz=−i
(z2
z2 +1
)= − 1
−i− i=− i
2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:
z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z+ i
1z− i
Resz=i
(z2
z2 +1
)= − 1
i+ i=
i2
Residue at −i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z− i
1z+ i
Resz=−i
(z2
z2 +1
)= − 1
−i− i=− i
2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:z2
z2 +1
=z2 +1−1
z2 +1= 1− 1
z2 +1= 1− 1
z+ i1
z− i
Resz=i
(z2
z2 +1
)= − 1
i+ i=
i2
Residue at −i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z− i
1z+ i
Resz=−i
(z2
z2 +1
)= − 1
−i− i=− i
2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z+ i
1z− i
Resz=i
(z2
z2 +1
)= − 1
i+ i=
i2
Residue at −i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z− i
1z+ i
Resz=−i
(z2
z2 +1
)= − 1
−i− i=− i
2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z+ i
1z− i
Resz=i
(z2
z2 +1
)= − 1
i+ i=
i2
Residue at −i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z− i
1z+ i
Resz=−i
(z2
z2 +1
)= − 1
−i− i=− i
2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z+ i
1z− i
Resz=i
(z2
z2 +1
)= − 1
i+ i=
i2
Residue at −i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z− i
1z+ i
Resz=−i
(z2
z2 +1
)= − 1
−i− i=− i
2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z+ i
1z− i
Resz=i
(z2
z2 +1
)
= − 1i+ i
=i2
Residue at −i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z− i
1z+ i
Resz=−i
(z2
z2 +1
)= − 1
−i− i=− i
2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z+ i
1z− i
Resz=i
(z2
z2 +1
)= − 1
i+ i
=i2
Residue at −i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z− i
1z+ i
Resz=−i
(z2
z2 +1
)= − 1
−i− i=− i
2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z+ i
1z− i
Resz=i
(z2
z2 +1
)= − 1
i+ i=
i2
Residue at −i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z− i
1z+ i
Resz=−i
(z2
z2 +1
)= − 1
−i− i=− i
2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z+ i
1z− i
Resz=i
(z2
z2 +1
)= − 1
i+ i=
i2
Residue at −i:
z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z− i
1z+ i
Resz=−i
(z2
z2 +1
)= − 1
−i− i=− i
2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z+ i
1z− i
Resz=i
(z2
z2 +1
)= − 1
i+ i=
i2
Residue at −i:z2
z2 +1
=z2 +1−1
z2 +1= 1− 1
z2 +1= 1− 1
z− i1
z+ i
Resz=−i
(z2
z2 +1
)= − 1
−i− i=− i
2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z+ i
1z− i
Resz=i
(z2
z2 +1
)= − 1
i+ i=
i2
Residue at −i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z− i
1z+ i
Resz=−i
(z2
z2 +1
)= − 1
−i− i=− i
2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z+ i
1z− i
Resz=i
(z2
z2 +1
)= − 1
i+ i=
i2
Residue at −i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z− i
1z+ i
Resz=−i
(z2
z2 +1
)= − 1
−i− i=− i
2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z+ i
1z− i
Resz=i
(z2
z2 +1
)= − 1
i+ i=
i2
Residue at −i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z− i
1z+ i
Resz=−i
(z2
z2 +1
)= − 1
−i− i=− i
2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z+ i
1z− i
Resz=i
(z2
z2 +1
)= − 1
i+ i=
i2
Residue at −i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z− i
1z+ i
Resz=−i
(z2
z2 +1
)
= − 1−i− i
=− i2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z+ i
1z− i
Resz=i
(z2
z2 +1
)= − 1
i+ i=
i2
Residue at −i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z− i
1z+ i
Resz=−i
(z2
z2 +1
)= − 1
−i− i
=− i2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
Residue at i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z+ i
1z− i
Resz=i
(z2
z2 +1
)= − 1
i+ i=
i2
Residue at −i:z2
z2 +1=
z2 +1−1z2 +1
= 1− 1z2 +1
= 1− 1z− i
1z+ i
Resz=−i
(z2
z2 +1
)= − 1
−i− i=− i
2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
∫C
z2
z2 +1dz = 2πi
[Resz=i
(z2
z2 +1
)+Resz=−i
(z2
z2 +1
)]= 2πi
[i2
+(− i
2
)]= 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
∫C
z2
z2 +1dz
= 2πi[
Resz=i
(z2
z2 +1
)+Resz=−i
(z2
z2 +1
)]= 2πi
[i2
+(− i
2
)]= 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
∫C
z2
z2 +1dz = 2πi
[Resz=i
(z2
z2 +1
)+Resz=−i
(z2
z2 +1
)]
= 2πi[
i2
+(− i
2
)]= 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
∫C
z2
z2 +1dz = 2πi
[Resz=i
(z2
z2 +1
)+Resz=−i
(z2
z2 +1
)]= 2πi
[i2
+(− i
2
)]
= 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
∫C
z2
z2 +1dz = 2πi
[Resz=i
(z2
z2 +1
)+Resz=−i
(z2
z2 +1
)]= 2πi
[i2
+(− i
2
)]= 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition.
Let f be analytic on C except for a finite number ofsingular points z1, . . . ,zn. Assume that R1 is so that all |zj| ≤ R1. ForR0 > R1 let C−0 be the circle around the origin of radius R0, traversedin the clockwise, that is, the mathematically negative direction. Thenwe define the residue at infinity of f as
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz.
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz =− 12πi
∫C+
0
f (z) dz =− 12πi
∫C+
0
∞
∑n=−∞
cnzn dz
= −c−1 =− 12πi
∫C+
0
∞
∑n=−∞
cn−2
zn dz =− 12πi
∫C+
0
1z2
∞
∑n=−∞
cn−2
zn−2 dz
= − 12πi
∫C+
0
1z2 f(
1z
)dz =−Resz=0
(1z2 f(
1z
))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let f be analytic on C except for a finite number ofsingular points z1, . . . ,zn.
Assume that R1 is so that all |zj| ≤ R1. ForR0 > R1 let C−0 be the circle around the origin of radius R0, traversedin the clockwise, that is, the mathematically negative direction. Thenwe define the residue at infinity of f as
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz.
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz =− 12πi
∫C+
0
f (z) dz =− 12πi
∫C+
0
∞
∑n=−∞
cnzn dz
= −c−1 =− 12πi
∫C+
0
∞
∑n=−∞
cn−2
zn dz =− 12πi
∫C+
0
1z2
∞
∑n=−∞
cn−2
zn−2 dz
= − 12πi
∫C+
0
1z2 f(
1z
)dz =−Resz=0
(1z2 f(
1z
))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let f be analytic on C except for a finite number ofsingular points z1, . . . ,zn. Assume that R1 is so that all |zj| ≤ R1.
ForR0 > R1 let C−0 be the circle around the origin of radius R0, traversedin the clockwise, that is, the mathematically negative direction. Thenwe define the residue at infinity of f as
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz.
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz =− 12πi
∫C+
0
f (z) dz =− 12πi
∫C+
0
∞
∑n=−∞
cnzn dz
= −c−1 =− 12πi
∫C+
0
∞
∑n=−∞
cn−2
zn dz =− 12πi
∫C+
0
1z2
∞
∑n=−∞
cn−2
zn−2 dz
= − 12πi
∫C+
0
1z2 f(
1z
)dz =−Resz=0
(1z2 f(
1z
))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let f be analytic on C except for a finite number ofsingular points z1, . . . ,zn. Assume that R1 is so that all |zj| ≤ R1. ForR0 > R1 let C−0 be the circle around the origin of radius R0, traversedin the clockwise
, that is, the mathematically negative direction. Thenwe define the residue at infinity of f as
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz.
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz =− 12πi
∫C+
0
f (z) dz =− 12πi
∫C+
0
∞
∑n=−∞
cnzn dz
= −c−1 =− 12πi
∫C+
0
∞
∑n=−∞
cn−2
zn dz =− 12πi
∫C+
0
1z2
∞
∑n=−∞
cn−2
zn−2 dz
= − 12πi
∫C+
0
1z2 f(
1z
)dz =−Resz=0
(1z2 f(
1z
))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let f be analytic on C except for a finite number ofsingular points z1, . . . ,zn. Assume that R1 is so that all |zj| ≤ R1. ForR0 > R1 let C−0 be the circle around the origin of radius R0, traversedin the clockwise, that is, the mathematically negative
direction. Thenwe define the residue at infinity of f as
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz.
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz =− 12πi
∫C+
0
f (z) dz =− 12πi
∫C+
0
∞
∑n=−∞
cnzn dz
= −c−1 =− 12πi
∫C+
0
∞
∑n=−∞
cn−2
zn dz =− 12πi
∫C+
0
1z2
∞
∑n=−∞
cn−2
zn−2 dz
= − 12πi
∫C+
0
1z2 f(
1z
)dz =−Resz=0
(1z2 f(
1z
))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let f be analytic on C except for a finite number ofsingular points z1, . . . ,zn. Assume that R1 is so that all |zj| ≤ R1. ForR0 > R1 let C−0 be the circle around the origin of radius R0, traversedin the clockwise, that is, the mathematically negative direction.
Thenwe define the residue at infinity of f as
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz.
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz =− 12πi
∫C+
0
f (z) dz =− 12πi
∫C+
0
∞
∑n=−∞
cnzn dz
= −c−1 =− 12πi
∫C+
0
∞
∑n=−∞
cn−2
zn dz =− 12πi
∫C+
0
1z2
∞
∑n=−∞
cn−2
zn−2 dz
= − 12πi
∫C+
0
1z2 f(
1z
)dz =−Resz=0
(1z2 f(
1z
))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let f be analytic on C except for a finite number ofsingular points z1, . . . ,zn. Assume that R1 is so that all |zj| ≤ R1. ForR0 > R1 let C−0 be the circle around the origin of radius R0, traversedin the clockwise, that is, the mathematically negative direction. Thenwe define the residue at infinity of f as
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz.
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz =− 12πi
∫C+
0
f (z) dz =− 12πi
∫C+
0
∞
∑n=−∞
cnzn dz
= −c−1 =− 12πi
∫C+
0
∞
∑n=−∞
cn−2
zn dz =− 12πi
∫C+
0
1z2
∞
∑n=−∞
cn−2
zn−2 dz
= − 12πi
∫C+
0
1z2 f(
1z
)dz =−Resz=0
(1z2 f(
1z
))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let f be analytic on C except for a finite number ofsingular points z1, . . . ,zn. Assume that R1 is so that all |zj| ≤ R1. ForR0 > R1 let C−0 be the circle around the origin of radius R0, traversedin the clockwise, that is, the mathematically negative direction. Thenwe define the residue at infinity of f as
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz.
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz
=− 12πi
∫C+
0
f (z) dz =− 12πi
∫C+
0
∞
∑n=−∞
cnzn dz
= −c−1 =− 12πi
∫C+
0
∞
∑n=−∞
cn−2
zn dz =− 12πi
∫C+
0
1z2
∞
∑n=−∞
cn−2
zn−2 dz
= − 12πi
∫C+
0
1z2 f(
1z
)dz =−Resz=0
(1z2 f(
1z
))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let f be analytic on C except for a finite number ofsingular points z1, . . . ,zn. Assume that R1 is so that all |zj| ≤ R1. ForR0 > R1 let C−0 be the circle around the origin of radius R0, traversedin the clockwise, that is, the mathematically negative direction. Thenwe define the residue at infinity of f as
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz.
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz =− 12πi
∫C+
0
f (z) dz
=− 12πi
∫C+
0
∞
∑n=−∞
cnzn dz
= −c−1 =− 12πi
∫C+
0
∞
∑n=−∞
cn−2
zn dz =− 12πi
∫C+
0
1z2
∞
∑n=−∞
cn−2
zn−2 dz
= − 12πi
∫C+
0
1z2 f(
1z
)dz =−Resz=0
(1z2 f(
1z
))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let f be analytic on C except for a finite number ofsingular points z1, . . . ,zn. Assume that R1 is so that all |zj| ≤ R1. ForR0 > R1 let C−0 be the circle around the origin of radius R0, traversedin the clockwise, that is, the mathematically negative direction. Thenwe define the residue at infinity of f as
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz.
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz =− 12πi
∫C+
0
f (z) dz =− 12πi
∫C+
0
∞
∑n=−∞
cnzn dz
= −c−1 =− 12πi
∫C+
0
∞
∑n=−∞
cn−2
zn dz =− 12πi
∫C+
0
1z2
∞
∑n=−∞
cn−2
zn−2 dz
= − 12πi
∫C+
0
1z2 f(
1z
)dz =−Resz=0
(1z2 f(
1z
))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let f be analytic on C except for a finite number ofsingular points z1, . . . ,zn. Assume that R1 is so that all |zj| ≤ R1. ForR0 > R1 let C−0 be the circle around the origin of radius R0, traversedin the clockwise, that is, the mathematically negative direction. Thenwe define the residue at infinity of f as
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz.
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz =− 12πi
∫C+
0
f (z) dz =− 12πi
∫C+
0
∞
∑n=−∞
cnzn dz
= −c−1
=− 12πi
∫C+
0
∞
∑n=−∞
cn−2
zn dz =− 12πi
∫C+
0
1z2
∞
∑n=−∞
cn−2
zn−2 dz
= − 12πi
∫C+
0
1z2 f(
1z
)dz =−Resz=0
(1z2 f(
1z
))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let f be analytic on C except for a finite number ofsingular points z1, . . . ,zn. Assume that R1 is so that all |zj| ≤ R1. ForR0 > R1 let C−0 be the circle around the origin of radius R0, traversedin the clockwise, that is, the mathematically negative direction. Thenwe define the residue at infinity of f as
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz.
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz =− 12πi
∫C+
0
f (z) dz =− 12πi
∫C+
0
∞
∑n=−∞
cnzn dz
= −c−1 =− 12πi
∫C+
0
∞
∑n=−∞
cn−2
zn dz
=− 12πi
∫C+
0
1z2
∞
∑n=−∞
cn−2
zn−2 dz
= − 12πi
∫C+
0
1z2 f(
1z
)dz =−Resz=0
(1z2 f(
1z
))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let f be analytic on C except for a finite number ofsingular points z1, . . . ,zn. Assume that R1 is so that all |zj| ≤ R1. ForR0 > R1 let C−0 be the circle around the origin of radius R0, traversedin the clockwise, that is, the mathematically negative direction. Thenwe define the residue at infinity of f as
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz.
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz =− 12πi
∫C+
0
f (z) dz =− 12πi
∫C+
0
∞
∑n=−∞
cnzn dz
= −c−1 =− 12πi
∫C+
0
∞
∑n=−∞
cn−2
zn dz =− 12πi
∫C+
0
1z2
∞
∑n=−∞
cn−2
zn−2 dz
= − 12πi
∫C+
0
1z2 f(
1z
)dz =−Resz=0
(1z2 f(
1z
))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let f be analytic on C except for a finite number ofsingular points z1, . . . ,zn. Assume that R1 is so that all |zj| ≤ R1. ForR0 > R1 let C−0 be the circle around the origin of radius R0, traversedin the clockwise, that is, the mathematically negative direction. Thenwe define the residue at infinity of f as
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz.
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz =− 12πi
∫C+
0
f (z) dz =− 12πi
∫C+
0
∞
∑n=−∞
cnzn dz
= −c−1 =− 12πi
∫C+
0
∞
∑n=−∞
cn−2
zn dz =− 12πi
∫C+
0
1z2
∞
∑n=−∞
cn−2
zn−2 dz
= − 12πi
∫C+
0
1z2 f(
1z
)dz
=−Resz=0
(1z2 f(
1z
))
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let f be analytic on C except for a finite number ofsingular points z1, . . . ,zn. Assume that R1 is so that all |zj| ≤ R1. ForR0 > R1 let C−0 be the circle around the origin of radius R0, traversedin the clockwise, that is, the mathematically negative direction. Thenwe define the residue at infinity of f as
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz.
Resz=∞(f ) =1
2πi
∫C−0
f (z) dz =− 12πi
∫C+
0
f (z) dz =− 12πi
∫C+
0
∞
∑n=−∞
cnzn dz
= −c−1 =− 12πi
∫C+
0
∞
∑n=−∞
cn−2
zn dz =− 12πi
∫C+
0
1z2
∞
∑n=−∞
cn−2
zn−2 dz
= − 12πi
∫C+
0
1z2 f(
1z
)dz =−Resz=0
(1z2 f(
1z
))Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem.
If f is analytic on C, except for a finite number of singularpoints that lie in the interior of a positively oriented simple closedcontour C, then ∫
Cf (z) dz = 2πiResz=0
[1z2 f(
1z
)].
Proof. See previous panel.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. If f is analytic on C, except for a finite number of singularpoints that lie in the interior of a positively oriented simple closedcontour C, then ∫
Cf (z) dz = 2πiResz=0
[1z2 f(
1z
)].
Proof. See previous panel.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. If f is analytic on C, except for a finite number of singularpoints that lie in the interior of a positively oriented simple closedcontour C, then ∫
Cf (z) dz = 2πiResz=0
[1z2 f(
1z
)].
Proof.
See previous panel.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. If f is analytic on C, except for a finite number of singularpoints that lie in the interior of a positively oriented simple closedcontour C, then ∫
Cf (z) dz = 2πiResz=0
[1z2 f(
1z
)].
Proof. See previous panel.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. If f is analytic on C, except for a finite number of singularpoints that lie in the interior of a positively oriented simple closedcontour C, then ∫
Cf (z) dz = 2πiResz=0
[1z2 f(
1z
)].
Proof. See previous panel.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example.
Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)
6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
r
i
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r
−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
-
6ℑ(z)
ℜ(z)6
�
?
-
3
ri
r−i
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
1z2 f(
1z
)=
1z2
(1z
)2(1z
)2+1
=1z2
11+ z2
=1z2
11− (−z2)
=1z2
∞
∑n=0
(−1)nz2n
=1z2 +
∞
∑n=1
(−1)nz2n−2
and hence∫
Cf (z) dz = 2πiResz=0
[1z2 f(
1z
)]= 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
1z2 f(
1z
)
=1z2
(1z
)2(1z
)2+1
=1z2
11+ z2
=1z2
11− (−z2)
=1z2
∞
∑n=0
(−1)nz2n
=1z2 +
∞
∑n=1
(−1)nz2n−2
and hence∫
Cf (z) dz = 2πiResz=0
[1z2 f(
1z
)]= 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
1z2 f(
1z
)=
1z2
(1z
)2(1z
)2+1
=1z2
11+ z2
=1z2
11− (−z2)
=1z2
∞
∑n=0
(−1)nz2n
=1z2 +
∞
∑n=1
(−1)nz2n−2
and hence∫
Cf (z) dz = 2πiResz=0
[1z2 f(
1z
)]= 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
1z2 f(
1z
)=
1z2
(1z
)2(1z
)2+1
=1z2
11+ z2
=1z2
11− (−z2)
=1z2
∞
∑n=0
(−1)nz2n
=1z2 +
∞
∑n=1
(−1)nz2n−2
and hence∫
Cf (z) dz = 2πiResz=0
[1z2 f(
1z
)]= 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
1z2 f(
1z
)=
1z2
(1z
)2(1z
)2+1
=1z2
11+ z2
=1z2
11− (−z2)
=1z2
∞
∑n=0
(−1)nz2n
=1z2 +
∞
∑n=1
(−1)nz2n−2
and hence∫
Cf (z) dz = 2πiResz=0
[1z2 f(
1z
)]= 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
1z2 f(
1z
)=
1z2
(1z
)2(1z
)2+1
=1z2
11+ z2
=1z2
11− (−z2)
=1z2
∞
∑n=0
(−1)nz2n
=1z2 +
∞
∑n=1
(−1)nz2n−2
and hence∫
Cf (z) dz = 2πiResz=0
[1z2 f(
1z
)]= 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
1z2 f(
1z
)=
1z2
(1z
)2(1z
)2+1
=1z2
11+ z2
=1z2
11− (−z2)
=1z2
∞
∑n=0
(−1)nz2n
=1z2 +
∞
∑n=1
(−1)nz2n−2
and hence∫
Cf (z) dz = 2πiResz=0
[1z2 f(
1z
)]= 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
1z2 f(
1z
)=
1z2
(1z
)2(1z
)2+1
=1z2
11+ z2
=1z2
11− (−z2)
=1z2
∞
∑n=0
(−1)nz2n
=1z2 +
∞
∑n=1
(−1)nz2n−2
and hence∫
Cf (z) dz
= 2πiResz=0
[1z2 f(
1z
)]= 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
1z2 f(
1z
)=
1z2
(1z
)2(1z
)2+1
=1z2
11+ z2
=1z2
11− (−z2)
=1z2
∞
∑n=0
(−1)nz2n
=1z2 +
∞
∑n=1
(−1)nz2n−2
and hence∫
Cf (z) dz = 2πiResz=0
[1z2 f(
1z
)]
= 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate the function f (z) =z2
z2 +1over the positively
oriented boundary of the square of numbers z = x+ iy with (x,y) in[−3,3]× [−3,3].
1z2 f(
1z
)=
1z2
(1z
)2(1z
)2+1
=1z2
11+ z2
=1z2
11− (−z2)
=1z2
∞
∑n=0
(−1)nz2n
=1z2 +
∞
∑n=1
(−1)nz2n−2
and hence∫
Cf (z) dz = 2πiResz=0
[1z2 f(
1z
)]= 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition.
Let z0 be a complex number, let r > 0 and let f be analyticin a deleted neighborhood of z0. In this case z0 is called an isolatedsingularity of f . Consider the Laurent expansion of f around z0
f (z) =∞
∑n=0
an(z− z0)n +b1
z− z0+
b2
(z− z0)2 +b3
(z− z0)3 + · · ·
The sum of the negative powers of z− z0 is also called the principalpart of f .
1. If bn = 0 for all n, then z0 is called a removable singularity.2. If there is a positive number m so that bm 6= 0 and bn = 0 for all
n > m, then z0 is called a pole of order m.3. If the number m in part 2 equals 1, then z0 is also called a
simple pole.4. If z0 is not removable and there is no number as in part 2, then z0
is called an essential singularity.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let z0 be a complex number, let r > 0 and let f be analyticin a deleted neighborhood of z0.
In this case z0 is called an isolatedsingularity of f . Consider the Laurent expansion of f around z0
f (z) =∞
∑n=0
an(z− z0)n +b1
z− z0+
b2
(z− z0)2 +b3
(z− z0)3 + · · ·
The sum of the negative powers of z− z0 is also called the principalpart of f .
1. If bn = 0 for all n, then z0 is called a removable singularity.2. If there is a positive number m so that bm 6= 0 and bn = 0 for all
n > m, then z0 is called a pole of order m.3. If the number m in part 2 equals 1, then z0 is also called a
simple pole.4. If z0 is not removable and there is no number as in part 2, then z0
is called an essential singularity.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let z0 be a complex number, let r > 0 and let f be analyticin a deleted neighborhood of z0. In this case z0 is called an isolatedsingularity of f .
Consider the Laurent expansion of f around z0
f (z) =∞
∑n=0
an(z− z0)n +b1
z− z0+
b2
(z− z0)2 +b3
(z− z0)3 + · · ·
The sum of the negative powers of z− z0 is also called the principalpart of f .
1. If bn = 0 for all n, then z0 is called a removable singularity.2. If there is a positive number m so that bm 6= 0 and bn = 0 for all
n > m, then z0 is called a pole of order m.3. If the number m in part 2 equals 1, then z0 is also called a
simple pole.4. If z0 is not removable and there is no number as in part 2, then z0
is called an essential singularity.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let z0 be a complex number, let r > 0 and let f be analyticin a deleted neighborhood of z0. In this case z0 is called an isolatedsingularity of f . Consider the Laurent expansion of f around z0
f (z) =∞
∑n=0
an(z− z0)n +b1
z− z0+
b2
(z− z0)2 +b3
(z− z0)3 + · · ·
The sum of the negative powers of z− z0 is also called the principalpart of f .
1. If bn = 0 for all n, then z0 is called a removable singularity.2. If there is a positive number m so that bm 6= 0 and bn = 0 for all
n > m, then z0 is called a pole of order m.3. If the number m in part 2 equals 1, then z0 is also called a
simple pole.4. If z0 is not removable and there is no number as in part 2, then z0
is called an essential singularity.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let z0 be a complex number, let r > 0 and let f be analyticin a deleted neighborhood of z0. In this case z0 is called an isolatedsingularity of f . Consider the Laurent expansion of f around z0
f (z) =∞
∑n=0
an(z− z0)n +b1
z− z0+
b2
(z− z0)2 +b3
(z− z0)3 + · · ·
The sum of the negative powers of z− z0 is also called the principalpart of f .
1. If bn = 0 for all n, then z0 is called a removable singularity.2. If there is a positive number m so that bm 6= 0 and bn = 0 for all
n > m, then z0 is called a pole of order m.3. If the number m in part 2 equals 1, then z0 is also called a
simple pole.4. If z0 is not removable and there is no number as in part 2, then z0
is called an essential singularity.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let z0 be a complex number, let r > 0 and let f be analyticin a deleted neighborhood of z0. In this case z0 is called an isolatedsingularity of f . Consider the Laurent expansion of f around z0
f (z) =∞
∑n=0
an(z− z0)n +b1
z− z0+
b2
(z− z0)2 +b3
(z− z0)3 + · · ·
The sum of the negative powers of z− z0 is also called the principalpart of f .
1. If bn = 0 for all n, then z0 is called a removable singularity.
2. If there is a positive number m so that bm 6= 0 and bn = 0 for alln > m, then z0 is called a pole of order m.
3. If the number m in part 2 equals 1, then z0 is also called asimple pole.
4. If z0 is not removable and there is no number as in part 2, then z0is called an essential singularity.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let z0 be a complex number, let r > 0 and let f be analyticin a deleted neighborhood of z0. In this case z0 is called an isolatedsingularity of f . Consider the Laurent expansion of f around z0
f (z) =∞
∑n=0
an(z− z0)n +b1
z− z0+
b2
(z− z0)2 +b3
(z− z0)3 + · · ·
The sum of the negative powers of z− z0 is also called the principalpart of f .
1. If bn = 0 for all n, then z0 is called a removable singularity.2. If there is a positive number m so that bm 6= 0 and bn = 0 for all
n > m, then z0 is called a pole of order m.
3. If the number m in part 2 equals 1, then z0 is also called asimple pole.
4. If z0 is not removable and there is no number as in part 2, then z0is called an essential singularity.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let z0 be a complex number, let r > 0 and let f be analyticin a deleted neighborhood of z0. In this case z0 is called an isolatedsingularity of f . Consider the Laurent expansion of f around z0
f (z) =∞
∑n=0
an(z− z0)n +b1
z− z0+
b2
(z− z0)2 +b3
(z− z0)3 + · · ·
The sum of the negative powers of z− z0 is also called the principalpart of f .
1. If bn = 0 for all n, then z0 is called a removable singularity.2. If there is a positive number m so that bm 6= 0 and bn = 0 for all
n > m, then z0 is called a pole of order m.3. If the number m in part 2 equals 1, then z0 is also called a
simple pole.
4. If z0 is not removable and there is no number as in part 2, then z0is called an essential singularity.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Definition. Let z0 be a complex number, let r > 0 and let f be analyticin a deleted neighborhood of z0. In this case z0 is called an isolatedsingularity of f . Consider the Laurent expansion of f around z0
f (z) =∞
∑n=0
an(z− z0)n +b1
z− z0+
b2
(z− z0)2 +b3
(z− z0)3 + · · ·
The sum of the negative powers of z− z0 is also called the principalpart of f .
1. If bn = 0 for all n, then z0 is called a removable singularity.2. If there is a positive number m so that bm 6= 0 and bn = 0 for all
n > m, then z0 is called a pole of order m.3. If the number m in part 2 equals 1, then z0 is also called a
simple pole.4. If z0 is not removable and there is no number as in part 2, then z0
is called an essential singularity.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example.
The function f (z) =sin(z)
zhas a removable singularity at
z = 0.sin(z)
z=
1z
sin(z)
=1z
∞
∑n=0
(−1)n
(2n+1)!z2n+1
=∞
∑n=0
(−1)n
(2n+1)!z2n
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =sin(z)
zhas a removable singularity at
z = 0.
sin(z)z
=1z
sin(z)
=1z
∞
∑n=0
(−1)n
(2n+1)!z2n+1
=∞
∑n=0
(−1)n
(2n+1)!z2n
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =sin(z)
zhas a removable singularity at
z = 0.sin(z)
z
=1z
sin(z)
=1z
∞
∑n=0
(−1)n
(2n+1)!z2n+1
=∞
∑n=0
(−1)n
(2n+1)!z2n
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =sin(z)
zhas a removable singularity at
z = 0.sin(z)
z=
1z
sin(z)
=1z
∞
∑n=0
(−1)n
(2n+1)!z2n+1
=∞
∑n=0
(−1)n
(2n+1)!z2n
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =sin(z)
zhas a removable singularity at
z = 0.sin(z)
z=
1z
sin(z)
=1z
∞
∑n=0
(−1)n
(2n+1)!z2n+1
=∞
∑n=0
(−1)n
(2n+1)!z2n
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =sin(z)
zhas a removable singularity at
z = 0.sin(z)
z=
1z
sin(z)
=1z
∞
∑n=0
(−1)n
(2n+1)!z2n+1
=∞
∑n=0
(−1)n
(2n+1)!z2n
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example.
The function f (z) =1
1− z2 has simple poles at z = 1 and at
z =−1.1
1− z2 =1
z+11
z−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
1− z2 has simple poles at z = 1 and at
z =−1.
11− z2 =
1z+1
1z−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
1− z2 has simple poles at z = 1 and at
z =−1.1
1− z2
=1
z+11
z−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1
1− z2 has simple poles at z = 1 and at
z =−1.1
1− z2 =1
z+11
z−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example.
The function f (z) =1+ z2
z3 has a pole of order 3 at z = 0.
1+ z2
z3 =1z3 +
1z
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1+ z2
z3 has a pole of order 3 at z = 0.
1+ z2
z3 =1z3 +
1z
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1+ z2
z3 has a pole of order 3 at z = 0.
1+ z2
z3
=1z3 +
1z
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) =1+ z2
z3 has a pole of order 3 at z = 0.
1+ z2
z3 =1z3 +
1z
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example.
The function f (z) = e1z has an essential singularity at
z = 0.
e1z =
∞
∑n=0
1n!
1zn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) = e1z has an essential singularity at
z = 0.
e1z =
∞
∑n=0
1n!
1zn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) = e1z has an essential singularity at
z = 0.
e1z
=∞
∑n=0
1n!
1zn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. The function f (z) = e1z has an essential singularity at
z = 0.
e1z =
∞
∑n=0
1n!
1zn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem.
Let the function f be analytic in a deleted neighborhood ofthe point z0. Then z0 is a pole of order m if and only if there is functionΦ that is analytic in a neighborhood of z0, Φ(z0) 6= 0 and so that
f (z) =Φ(z)
(z− z0)m
for all z in a deleted neighborhood of z0. Moreover,
Resz=z0(f ) =Φ(m−1)(z0)(m−1)!
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. Let the function f be analytic in a deleted neighborhood ofthe point z0.
Then z0 is a pole of order m if and only if there is functionΦ that is analytic in a neighborhood of z0, Φ(z0) 6= 0 and so that
f (z) =Φ(z)
(z− z0)m
for all z in a deleted neighborhood of z0. Moreover,
Resz=z0(f ) =Φ(m−1)(z0)(m−1)!
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. Let the function f be analytic in a deleted neighborhood ofthe point z0. Then z0 is a pole of order m if and only if there is functionΦ that is analytic in a neighborhood of z0
, Φ(z0) 6= 0 and so that
f (z) =Φ(z)
(z− z0)m
for all z in a deleted neighborhood of z0. Moreover,
Resz=z0(f ) =Φ(m−1)(z0)(m−1)!
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. Let the function f be analytic in a deleted neighborhood ofthe point z0. Then z0 is a pole of order m if and only if there is functionΦ that is analytic in a neighborhood of z0, Φ(z0) 6= 0
and so that
f (z) =Φ(z)
(z− z0)m
for all z in a deleted neighborhood of z0. Moreover,
Resz=z0(f ) =Φ(m−1)(z0)(m−1)!
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. Let the function f be analytic in a deleted neighborhood ofthe point z0. Then z0 is a pole of order m if and only if there is functionΦ that is analytic in a neighborhood of z0, Φ(z0) 6= 0 and so that
f (z) =Φ(z)
(z− z0)m
for all z in a deleted neighborhood of z0.
Moreover,
Resz=z0(f ) =Φ(m−1)(z0)(m−1)!
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Theorem. Let the function f be analytic in a deleted neighborhood ofthe point z0. Then z0 is a pole of order m if and only if there is functionΦ that is analytic in a neighborhood of z0, Φ(z0) 6= 0 and so that
f (z) =Φ(z)
(z− z0)m
for all z in a deleted neighborhood of z0. Moreover,
Resz=z0(f ) =Φ(m−1)(z0)(m−1)!
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof.
If f has a pole of order m around z0, then the Laurentexpansion of z for 0 < |z− z0|< R (for some R) is (with c−m 6= 0)
f (z) =∞
∑n=−m
cn(z− z0)n =1
(z− z0)m
∞
∑n=−m
cn(z− z0)n+m
=1
(z− z0)m
∞
∑k=0
ck−m(z− z0)k =:Φ(z)
(z− z0)m .
Conversely, if f (z) =Φ(z)
(z− z0)m with Φ(z0) 6= 0, then the above
computation in reverse shows that f has a pole of order m. Moreover,in this situation, the coefficient c−1 = Resz=z0(f ) of the Laurent
expansion of f is the coefficient am−1 =Φ(m−1)(z0)(m−1)!
of the power
series expansion of Φ.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof. If f has a pole of order m around z0, then the Laurentexpansion of z for 0 < |z− z0|< R (for some R) is (with c−m 6= 0)
f (z) =∞
∑n=−m
cn(z− z0)n
=1
(z− z0)m
∞
∑n=−m
cn(z− z0)n+m
=1
(z− z0)m
∞
∑k=0
ck−m(z− z0)k =:Φ(z)
(z− z0)m .
Conversely, if f (z) =Φ(z)
(z− z0)m with Φ(z0) 6= 0, then the above
computation in reverse shows that f has a pole of order m. Moreover,in this situation, the coefficient c−1 = Resz=z0(f ) of the Laurent
expansion of f is the coefficient am−1 =Φ(m−1)(z0)(m−1)!
of the power
series expansion of Φ.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof. If f has a pole of order m around z0, then the Laurentexpansion of z for 0 < |z− z0|< R (for some R) is (with c−m 6= 0)
f (z) =∞
∑n=−m
cn(z− z0)n =1
(z− z0)m
∞
∑n=−m
cn(z− z0)n+m
=1
(z− z0)m
∞
∑k=0
ck−m(z− z0)k =:Φ(z)
(z− z0)m .
Conversely, if f (z) =Φ(z)
(z− z0)m with Φ(z0) 6= 0, then the above
computation in reverse shows that f has a pole of order m. Moreover,in this situation, the coefficient c−1 = Resz=z0(f ) of the Laurent
expansion of f is the coefficient am−1 =Φ(m−1)(z0)(m−1)!
of the power
series expansion of Φ.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof. If f has a pole of order m around z0, then the Laurentexpansion of z for 0 < |z− z0|< R (for some R) is (with c−m 6= 0)
f (z) =∞
∑n=−m
cn(z− z0)n =1
(z− z0)m
∞
∑n=−m
cn(z− z0)n+m
=1
(z− z0)m
∞
∑k=0
ck−m(z− z0)k
=:Φ(z)
(z− z0)m .
Conversely, if f (z) =Φ(z)
(z− z0)m with Φ(z0) 6= 0, then the above
computation in reverse shows that f has a pole of order m. Moreover,in this situation, the coefficient c−1 = Resz=z0(f ) of the Laurent
expansion of f is the coefficient am−1 =Φ(m−1)(z0)(m−1)!
of the power
series expansion of Φ.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof. If f has a pole of order m around z0, then the Laurentexpansion of z for 0 < |z− z0|< R (for some R) is (with c−m 6= 0)
f (z) =∞
∑n=−m
cn(z− z0)n =1
(z− z0)m
∞
∑n=−m
cn(z− z0)n+m
=1
(z− z0)m
∞
∑k=0
ck−m(z− z0)k =:Φ(z)
(z− z0)m .
Conversely, if f (z) =Φ(z)
(z− z0)m with Φ(z0) 6= 0, then the above
computation in reverse shows that f has a pole of order m. Moreover,in this situation, the coefficient c−1 = Resz=z0(f ) of the Laurent
expansion of f is the coefficient am−1 =Φ(m−1)(z0)(m−1)!
of the power
series expansion of Φ.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof. If f has a pole of order m around z0, then the Laurentexpansion of z for 0 < |z− z0|< R (for some R) is (with c−m 6= 0)
f (z) =∞
∑n=−m
cn(z− z0)n =1
(z− z0)m
∞
∑n=−m
cn(z− z0)n+m
=1
(z− z0)m
∞
∑k=0
ck−m(z− z0)k =:Φ(z)
(z− z0)m .
Conversely, if f (z) =Φ(z)
(z− z0)m with Φ(z0) 6= 0, then the above
computation in reverse shows that f has a pole of order m.
Moreover,in this situation, the coefficient c−1 = Resz=z0(f ) of the Laurent
expansion of f is the coefficient am−1 =Φ(m−1)(z0)(m−1)!
of the power
series expansion of Φ.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof. If f has a pole of order m around z0, then the Laurentexpansion of z for 0 < |z− z0|< R (for some R) is (with c−m 6= 0)
f (z) =∞
∑n=−m
cn(z− z0)n =1
(z− z0)m
∞
∑n=−m
cn(z− z0)n+m
=1
(z− z0)m
∞
∑k=0
ck−m(z− z0)k =:Φ(z)
(z− z0)m .
Conversely, if f (z) =Φ(z)
(z− z0)m with Φ(z0) 6= 0, then the above
computation in reverse shows that f has a pole of order m. Moreover,in this situation, the coefficient c−1 = Resz=z0(f ) of the Laurent
expansion of f
is the coefficient am−1 =Φ(m−1)(z0)(m−1)!
of the power
series expansion of Φ.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof. If f has a pole of order m around z0, then the Laurentexpansion of z for 0 < |z− z0|< R (for some R) is (with c−m 6= 0)
f (z) =∞
∑n=−m
cn(z− z0)n =1
(z− z0)m
∞
∑n=−m
cn(z− z0)n+m
=1
(z− z0)m
∞
∑k=0
ck−m(z− z0)k =:Φ(z)
(z− z0)m .
Conversely, if f (z) =Φ(z)
(z− z0)m with Φ(z0) 6= 0, then the above
computation in reverse shows that f has a pole of order m. Moreover,in this situation, the coefficient c−1 = Resz=z0(f ) of the Laurent
expansion of f is the coefficient am−1 =Φ(m−1)(z0)(m−1)!
of the power
series expansion of Φ.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
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Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Proof. If f has a pole of order m around z0, then the Laurentexpansion of z for 0 < |z− z0|< R (for some R) is (with c−m 6= 0)
f (z) =∞
∑n=−m
cn(z− z0)n =1
(z− z0)m
∞
∑n=−m
cn(z− z0)n+m
=1
(z− z0)m
∞
∑k=0
ck−m(z− z0)k =:Φ(z)
(z− z0)m .
Conversely, if f (z) =Φ(z)
(z− z0)m with Φ(z0) 6= 0, then the above
computation in reverse shows that f has a pole of order m. Moreover,in this situation, the coefficient c−1 = Resz=z0(f ) of the Laurent
expansion of f is the coefficient am−1 =Φ(m−1)(z0)(m−1)!
of the power
series expansion of Φ.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example.
Find all poles, their order and their residues for
f (z) =(
z+1z−1
)3
.
The only singularity of f is z = 1.
f (z) =(
z+1z−1
)3
=1
(z−1)3
(z3 +3z2 +3z+1
)Φ(z) = z3 +3z2 +3z+1, m = 3
Φ′(z) = 3z2 +6z+3
Φ′′(z) = 6z+6
Φ′′(1) = 12
Resz=1(f ) =122!
= 6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Find all poles, their order and their residues for
f (z) =(
z+1z−1
)3
.
The only singularity of f is z = 1.
f (z) =(
z+1z−1
)3
=1
(z−1)3
(z3 +3z2 +3z+1
)Φ(z) = z3 +3z2 +3z+1, m = 3
Φ′(z) = 3z2 +6z+3
Φ′′(z) = 6z+6
Φ′′(1) = 12
Resz=1(f ) =122!
= 6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Find all poles, their order and their residues for
f (z) =(
z+1z−1
)3
.
The only singularity of f is z = 1.
f (z) =(
z+1z−1
)3
=1
(z−1)3
(z3 +3z2 +3z+1
)Φ(z) = z3 +3z2 +3z+1, m = 3
Φ′(z) = 3z2 +6z+3
Φ′′(z) = 6z+6
Φ′′(1) = 12
Resz=1(f ) =122!
= 6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Find all poles, their order and their residues for
f (z) =(
z+1z−1
)3
.
The only singularity of f is z = 1.
f (z) =(
z+1z−1
)3
=1
(z−1)3
(z3 +3z2 +3z+1
)Φ(z) = z3 +3z2 +3z+1, m = 3
Φ′(z) = 3z2 +6z+3
Φ′′(z) = 6z+6
Φ′′(1) = 12
Resz=1(f ) =122!
= 6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Find all poles, their order and their residues for
f (z) =(
z+1z−1
)3
.
The only singularity of f is z = 1.
f (z) =(
z+1z−1
)3
=1
(z−1)3
(z3 +3z2 +3z+1
)
Φ(z) = z3 +3z2 +3z+1, m = 3
Φ′(z) = 3z2 +6z+3
Φ′′(z) = 6z+6
Φ′′(1) = 12
Resz=1(f ) =122!
= 6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Find all poles, their order and their residues for
f (z) =(
z+1z−1
)3
.
The only singularity of f is z = 1.
f (z) =(
z+1z−1
)3
=1
(z−1)3
(z3 +3z2 +3z+1
)Φ(z) = z3 +3z2 +3z+1
, m = 3
Φ′(z) = 3z2 +6z+3
Φ′′(z) = 6z+6
Φ′′(1) = 12
Resz=1(f ) =122!
= 6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Find all poles, their order and their residues for
f (z) =(
z+1z−1
)3
.
The only singularity of f is z = 1.
f (z) =(
z+1z−1
)3
=1
(z−1)3
(z3 +3z2 +3z+1
)Φ(z) = z3 +3z2 +3z+1, m = 3
Φ′(z) = 3z2 +6z+3
Φ′′(z) = 6z+6
Φ′′(1) = 12
Resz=1(f ) =122!
= 6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Find all poles, their order and their residues for
f (z) =(
z+1z−1
)3
.
The only singularity of f is z = 1.
f (z) =(
z+1z−1
)3
=1
(z−1)3
(z3 +3z2 +3z+1
)Φ(z) = z3 +3z2 +3z+1, m = 3
Φ′(z) = 3z2 +6z+3
Φ′′(z) = 6z+6
Φ′′(1) = 12
Resz=1(f ) =122!
= 6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Find all poles, their order and their residues for
f (z) =(
z+1z−1
)3
.
The only singularity of f is z = 1.
f (z) =(
z+1z−1
)3
=1
(z−1)3
(z3 +3z2 +3z+1
)Φ(z) = z3 +3z2 +3z+1, m = 3
Φ′(z) = 3z2 +6z+3
Φ′′(z) = 6z+6
Φ′′(1) = 12
Resz=1(f ) =122!
= 6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Find all poles, their order and their residues for
f (z) =(
z+1z−1
)3
.
The only singularity of f is z = 1.
f (z) =(
z+1z−1
)3
=1
(z−1)3
(z3 +3z2 +3z+1
)Φ(z) = z3 +3z2 +3z+1, m = 3
Φ′(z) = 3z2 +6z+3
Φ′′(z) = 6z+6
Φ′′(1) = 12
Resz=1(f ) =122!
= 6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Find all poles, their order and their residues for
f (z) =(
z+1z−1
)3
.
The only singularity of f is z = 1.
f (z) =(
z+1z−1
)3
=1
(z−1)3
(z3 +3z2 +3z+1
)Φ(z) = z3 +3z2 +3z+1, m = 3
Φ′(z) = 3z2 +6z+3
Φ′′(z) = 6z+6
Φ′′(1) = 12
Resz=1(f ) =122!
= 6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Find all poles, their order and their residues for
f (z) =(
z+1z−1
)3
.
The only singularity of f is z = 1.
f (z) =(
z+1z−1
)3
=1
(z−1)3
(z3 +3z2 +3z+1
)Φ(z) = z3 +3z2 +3z+1, m = 3
Φ′(z) = 3z2 +6z+3
Φ′′(z) = 6z+6
Φ′′(1) = 12
Resz=1(f ) =122!
= 6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example.
Show that Resz=iLog(z)
(z2 +1)2 =π +2i
8.
Log(z)
(z2 +1)2 =Log(z)(
(z+ i)(z− i))2 =
1(z− i)2
Log(z)(z+ i)2
Φ(z) =Log(z)(z+ i)2 , m = 2
Φ′(z) =
1z (z+ i)2−2(z+ i)Log(z)
(z+ i)4 =1z (z+ i)−2Log(z)
(z+ i)3
=z+ i−2Log(z)z
z(z+ i)3
Φ′(i) =
i+ i−2Log(i)ii(i+ i)3 =
2i−2i π
2 ii(2i)3 =
2i+π
8
Resz=iLog(z)
(z2 +1)2 =11!
π +2i8
=π +2i
8
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Show that Resz=iLog(z)
(z2 +1)2 =π +2i
8.
Log(z)
(z2 +1)2 =Log(z)(
(z+ i)(z− i))2 =
1(z− i)2
Log(z)(z+ i)2
Φ(z) =Log(z)(z+ i)2 , m = 2
Φ′(z) =
1z (z+ i)2−2(z+ i)Log(z)
(z+ i)4 =1z (z+ i)−2Log(z)
(z+ i)3
=z+ i−2Log(z)z
z(z+ i)3
Φ′(i) =
i+ i−2Log(i)ii(i+ i)3 =
2i−2i π
2 ii(2i)3 =
2i+π
8
Resz=iLog(z)
(z2 +1)2 =11!
π +2i8
=π +2i
8
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Show that Resz=iLog(z)
(z2 +1)2 =π +2i
8.
Log(z)
(z2 +1)2 =Log(z)(
(z+ i)(z− i))2
=1
(z− i)2Log(z)(z+ i)2
Φ(z) =Log(z)(z+ i)2 , m = 2
Φ′(z) =
1z (z+ i)2−2(z+ i)Log(z)
(z+ i)4 =1z (z+ i)−2Log(z)
(z+ i)3
=z+ i−2Log(z)z
z(z+ i)3
Φ′(i) =
i+ i−2Log(i)ii(i+ i)3 =
2i−2i π
2 ii(2i)3 =
2i+π
8
Resz=iLog(z)
(z2 +1)2 =11!
π +2i8
=π +2i
8
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Show that Resz=iLog(z)
(z2 +1)2 =π +2i
8.
Log(z)
(z2 +1)2 =Log(z)(
(z+ i)(z− i))2 =
1(z− i)2
Log(z)(z+ i)2
Φ(z) =Log(z)(z+ i)2 , m = 2
Φ′(z) =
1z (z+ i)2−2(z+ i)Log(z)
(z+ i)4 =1z (z+ i)−2Log(z)
(z+ i)3
=z+ i−2Log(z)z
z(z+ i)3
Φ′(i) =
i+ i−2Log(i)ii(i+ i)3 =
2i−2i π
2 ii(2i)3 =
2i+π
8
Resz=iLog(z)
(z2 +1)2 =11!
π +2i8
=π +2i
8
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Show that Resz=iLog(z)
(z2 +1)2 =π +2i
8.
Log(z)
(z2 +1)2 =Log(z)(
(z+ i)(z− i))2 =
1(z− i)2
Log(z)(z+ i)2
Φ(z) =Log(z)(z+ i)2
, m = 2
Φ′(z) =
1z (z+ i)2−2(z+ i)Log(z)
(z+ i)4 =1z (z+ i)−2Log(z)
(z+ i)3
=z+ i−2Log(z)z
z(z+ i)3
Φ′(i) =
i+ i−2Log(i)ii(i+ i)3 =
2i−2i π
2 ii(2i)3 =
2i+π
8
Resz=iLog(z)
(z2 +1)2 =11!
π +2i8
=π +2i
8
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Show that Resz=iLog(z)
(z2 +1)2 =π +2i
8.
Log(z)
(z2 +1)2 =Log(z)(
(z+ i)(z− i))2 =
1(z− i)2
Log(z)(z+ i)2
Φ(z) =Log(z)(z+ i)2 , m = 2
Φ′(z) =
1z (z+ i)2−2(z+ i)Log(z)
(z+ i)4 =1z (z+ i)−2Log(z)
(z+ i)3
=z+ i−2Log(z)z
z(z+ i)3
Φ′(i) =
i+ i−2Log(i)ii(i+ i)3 =
2i−2i π
2 ii(2i)3 =
2i+π
8
Resz=iLog(z)
(z2 +1)2 =11!
π +2i8
=π +2i
8
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Show that Resz=iLog(z)
(z2 +1)2 =π +2i
8.
Log(z)
(z2 +1)2 =Log(z)(
(z+ i)(z− i))2 =
1(z− i)2
Log(z)(z+ i)2
Φ(z) =Log(z)(z+ i)2 , m = 2
Φ′(z) =
1z (z+ i)2−2(z+ i)Log(z)
(z+ i)4
=1z (z+ i)−2Log(z)
(z+ i)3
=z+ i−2Log(z)z
z(z+ i)3
Φ′(i) =
i+ i−2Log(i)ii(i+ i)3 =
2i−2i π
2 ii(2i)3 =
2i+π
8
Resz=iLog(z)
(z2 +1)2 =11!
π +2i8
=π +2i
8
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Show that Resz=iLog(z)
(z2 +1)2 =π +2i
8.
Log(z)
(z2 +1)2 =Log(z)(
(z+ i)(z− i))2 =
1(z− i)2
Log(z)(z+ i)2
Φ(z) =Log(z)(z+ i)2 , m = 2
Φ′(z) =
1z (z+ i)2−2(z+ i)Log(z)
(z+ i)4 =1z (z+ i)−2Log(z)
(z+ i)3
=z+ i−2Log(z)z
z(z+ i)3
Φ′(i) =
i+ i−2Log(i)ii(i+ i)3 =
2i−2i π
2 ii(2i)3 =
2i+π
8
Resz=iLog(z)
(z2 +1)2 =11!
π +2i8
=π +2i
8
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Show that Resz=iLog(z)
(z2 +1)2 =π +2i
8.
Log(z)
(z2 +1)2 =Log(z)(
(z+ i)(z− i))2 =
1(z− i)2
Log(z)(z+ i)2
Φ(z) =Log(z)(z+ i)2 , m = 2
Φ′(z) =
1z (z+ i)2−2(z+ i)Log(z)
(z+ i)4 =1z (z+ i)−2Log(z)
(z+ i)3
=z+ i−2Log(z)z
z(z+ i)3
Φ′(i) =
i+ i−2Log(i)ii(i+ i)3 =
2i−2i π
2 ii(2i)3 =
2i+π
8
Resz=iLog(z)
(z2 +1)2 =11!
π +2i8
=π +2i
8
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Show that Resz=iLog(z)
(z2 +1)2 =π +2i
8.
Log(z)
(z2 +1)2 =Log(z)(
(z+ i)(z− i))2 =
1(z− i)2
Log(z)(z+ i)2
Φ(z) =Log(z)(z+ i)2 , m = 2
Φ′(z) =
1z (z+ i)2−2(z+ i)Log(z)
(z+ i)4 =1z (z+ i)−2Log(z)
(z+ i)3
=z+ i−2Log(z)z
z(z+ i)3
Φ′(i) =
i+ i−2Log(i)ii(i+ i)3
=2i−2i π
2 ii(2i)3 =
2i+π
8
Resz=iLog(z)
(z2 +1)2 =11!
π +2i8
=π +2i
8
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Show that Resz=iLog(z)
(z2 +1)2 =π +2i
8.
Log(z)
(z2 +1)2 =Log(z)(
(z+ i)(z− i))2 =
1(z− i)2
Log(z)(z+ i)2
Φ(z) =Log(z)(z+ i)2 , m = 2
Φ′(z) =
1z (z+ i)2−2(z+ i)Log(z)
(z+ i)4 =1z (z+ i)−2Log(z)
(z+ i)3
=z+ i−2Log(z)z
z(z+ i)3
Φ′(i) =
i+ i−2Log(i)ii(i+ i)3 =
2i−2i π
2 ii(2i)3
=2i+π
8
Resz=iLog(z)
(z2 +1)2 =11!
π +2i8
=π +2i
8
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Show that Resz=iLog(z)
(z2 +1)2 =π +2i
8.
Log(z)
(z2 +1)2 =Log(z)(
(z+ i)(z− i))2 =
1(z− i)2
Log(z)(z+ i)2
Φ(z) =Log(z)(z+ i)2 , m = 2
Φ′(z) =
1z (z+ i)2−2(z+ i)Log(z)
(z+ i)4 =1z (z+ i)−2Log(z)
(z+ i)3
=z+ i−2Log(z)z
z(z+ i)3
Φ′(i) =
i+ i−2Log(i)ii(i+ i)3 =
2i−2i π
2 ii(2i)3 =
2i+π
8
Resz=iLog(z)
(z2 +1)2 =11!
π +2i8
=π +2i
8
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Show that Resz=iLog(z)
(z2 +1)2 =π +2i
8.
Log(z)
(z2 +1)2 =Log(z)(
(z+ i)(z− i))2 =
1(z− i)2
Log(z)(z+ i)2
Φ(z) =Log(z)(z+ i)2 , m = 2
Φ′(z) =
1z (z+ i)2−2(z+ i)Log(z)
(z+ i)4 =1z (z+ i)−2Log(z)
(z+ i)3
=z+ i−2Log(z)z
z(z+ i)3
Φ′(i) =
i+ i−2Log(i)ii(i+ i)3 =
2i−2i π
2 ii(2i)3 =
2i+π
8
Resz=iLog(z)
(z2 +1)2 =11!
π +2i8
=π +2i
8
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Show that Resz=iLog(z)
(z2 +1)2 =π +2i
8.
Log(z)
(z2 +1)2 =Log(z)(
(z+ i)(z− i))2 =
1(z− i)2
Log(z)(z+ i)2
Φ(z) =Log(z)(z+ i)2 , m = 2
Φ′(z) =
1z (z+ i)2−2(z+ i)Log(z)
(z+ i)4 =1z (z+ i)−2Log(z)
(z+ i)3
=z+ i−2Log(z)z
z(z+ i)3
Φ′(i) =
i+ i−2Log(i)ii(i+ i)3 =
2i−2i π
2 ii(2i)3 =
2i+π
8
Resz=iLog(z)
(z2 +1)2 =11!
π +2i8
=π +2i
8
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example.
Integrate f (z) =1
z3(z+4)around the positively oriented
circle of radius 5 around the origin.
We will need the residues of f at 0 and at −4, or we need the residue
at 0 of1z2 f(
1z
).
1z2 f(
1z
)=
1z2
1(1z
)3 (1z +4
) =1
1z
1+4zz
=z2
1+4z
As1z2 f(
1z
)does not have a singularity at 0, the integral must be 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate f (z) =1
z3(z+4)around the positively oriented
circle of radius 5 around the origin.
We will need the residues of f at 0 and at −4, or we need the residue
at 0 of1z2 f(
1z
).
1z2 f(
1z
)=
1z2
1(1z
)3 (1z +4
) =1
1z
1+4zz
=z2
1+4z
As1z2 f(
1z
)does not have a singularity at 0, the integral must be 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate f (z) =1
z3(z+4)around the positively oriented
circle of radius 5 around the origin.
We will need the residues of f at 0 and at −4
, or we need the residue
at 0 of1z2 f(
1z
).
1z2 f(
1z
)=
1z2
1(1z
)3 (1z +4
) =1
1z
1+4zz
=z2
1+4z
As1z2 f(
1z
)does not have a singularity at 0, the integral must be 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate f (z) =1
z3(z+4)around the positively oriented
circle of radius 5 around the origin.
We will need the residues of f at 0 and at −4, or we need the residue
at 0 of1z2 f(
1z
).
1z2 f(
1z
)=
1z2
1(1z
)3 (1z +4
) =1
1z
1+4zz
=z2
1+4z
As1z2 f(
1z
)does not have a singularity at 0, the integral must be 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate f (z) =1
z3(z+4)around the positively oriented
circle of radius 5 around the origin.
We will need the residues of f at 0 and at −4, or we need the residue
at 0 of1z2 f(
1z
).
1z2 f(
1z
)
=1z2
1(1z
)3 (1z +4
) =1
1z
1+4zz
=z2
1+4z
As1z2 f(
1z
)does not have a singularity at 0, the integral must be 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate f (z) =1
z3(z+4)around the positively oriented
circle of radius 5 around the origin.
We will need the residues of f at 0 and at −4, or we need the residue
at 0 of1z2 f(
1z
).
1z2 f(
1z
)=
1z2
1(1z
)3 (1z +4
)
=1
1z
1+4zz
=z2
1+4z
As1z2 f(
1z
)does not have a singularity at 0, the integral must be 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate f (z) =1
z3(z+4)around the positively oriented
circle of radius 5 around the origin.
We will need the residues of f at 0 and at −4, or we need the residue
at 0 of1z2 f(
1z
).
1z2 f(
1z
)=
1z2
1(1z
)3 (1z +4
) =1
1z
1+4zz
=z2
1+4z
As1z2 f(
1z
)does not have a singularity at 0, the integral must be 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate f (z) =1
z3(z+4)around the positively oriented
circle of radius 5 around the origin.
We will need the residues of f at 0 and at −4, or we need the residue
at 0 of1z2 f(
1z
).
1z2 f(
1z
)=
1z2
1(1z
)3 (1z +4
) =1
1z
1+4zz
=z2
1+4z
As1z2 f(
1z
)does not have a singularity at 0, the integral must be 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem
logo1
Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles
Example. Integrate f (z) =1
z3(z+4)around the positively oriented
circle of radius 5 around the origin.
We will need the residues of f at 0 and at −4, or we need the residue
at 0 of1z2 f(
1z
).
1z2 f(
1z
)=
1z2
1(1z
)3 (1z +4
) =1
1z
1+4zz
=z2
1+4z
As1z2 f(
1z
)does not have a singularity at 0, the integral must be 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Residue Theorem