complex integration & variables
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Complex Integration& VariablesAvradip Ghosh
University of Houston
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Complex Integration
We often interpret real integrals in terms of area; now we define complex integrals in terms ofline integrals over paths in the complex plane.
Reference: Mathematical Methods for Physics and Engineering. K.F. Riley ,M.P. Hobson, S.J. Bence, Cambridge University Press
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Reference: Mathematical Methods for Physics and Engineering. K.F. Riley ,M.P. Hobson, S.J. Bence, Cambridge University Press
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Reference: Mathematical Methods for Physics and Engineering. K.F. Riley ,M.P. Hobson, S.J. Bence, Cambridge University Press
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An Important Result
M is an upper bound on thevalue of |f(z)| on the path, i.e. |f(z)| ≤ M on C
L is the length of the path C
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What Is an Analytic Function
A function that is single-valued and differentiable at all points of a domain Ris said to be analytic (or regular) in R.
A function may be analytic in a domain except at a finite number of points (or an infinite number if the domain isinfinite);
Such Finite points are called
Singularities
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Cauchy-Riemann Relations (to check for analyticity)
real imaginary
If we have a function that is
divided into its real and imaginary parts
to check for analyticity (differentiability)
Necessary and sufficient condition is that the four partial derivativesexist, are continuous and satisfy the Cauchy–Riemann relations
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Reference: Mathematical Methods for Physics and Engineering. K.F. Riley ,M.P. Hobson, S.J. Bence, Cambridge University Press
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Power Series and Convergence
Convergence meaning:
The sum of the series either goes
to 0 or a finite value as the
terms goes to ∞
substituting
Converges if: Is also convergent
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Radius and Circle of Convergence
R
|z|<R… Convergent|z|>R… Divergent
|z|=R… Inconclusive
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Laurent Series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes even
terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.
𝑓(𝑧) =
𝑛=−∞
∞
𝑎𝑛 𝑧 − 𝑐 𝑛
𝑎𝑛 =1
2𝜋𝑖
𝛾
)𝑓(𝑧
𝑧 − 𝑐 𝑛+1 𝑑𝑧Nth Coefficient :
𝑓(𝑧) =)𝑓0(𝑧
𝑧 − 𝑧0𝑛=
𝑎𝑛𝑧 − 𝑧0
𝑛+. . . . . . . .
𝑎−1𝑧 − 𝑧0
+ 𝑎0 + 𝑎1(𝑧 − 𝑧0) + 𝑎1 𝑧 − 𝑧02+. . . .
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Singularities, Zeros, and Poles
the point z=α is called a singular point, or singularity of the complex function f(z) if f is not analytic at z=α but every neighbourhood of it
If a function f(z) is not analytic at z=αbut analytic everywhere in the entire
plane the point z=α is an Isolated Singularity of the function f(z)
Isolated Singularity:
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Types of Singularities
𝑓(𝑧) =1
𝑧 − 1 2
Pole of order 2 at z=1
𝑓(𝑧) = 𝑒 Τ1 𝑥 = 1 +1
𝑥+
1
2𝑥2+
1
6𝑥3
Removable Essential Pole
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Zeros of a Complex Function
Reference: Mathematical Methods for Physics and Engineering. K.F. Riley ,M.P. Hobson, S.J. Bence, Cambridge University Press
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Cauchy’s Theorem
Note: For Proof Refer to Riley Hobson
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Cauchy’s Residue Theorem
If we have a closed contour ׯ𝐶 and there
are poles inside the contour then we can close the contour around the poles
(only poles that are inside the contour)
Take the residue at the selected poles
Select the poles according to the boundary conditions and
the specific problem
𝑓(𝑧) =1
𝑧 + 𝑧1)(𝑧 − 𝑧2
Z1,Z2>0
Most Important
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Calculating Residue’s
For a pole of Order 1
𝑓(𝑧) =)𝑓0(𝑧
𝑧 − 𝑧0=
𝑎−1𝑧 − 𝑧0
+ 𝑎0 + 𝑎1(𝑧 − 𝑧0) + 𝑎1 𝑧 − 𝑧02+. . . .
Residue
Re𝑠𝑖𝑑𝑢𝑒|𝑃𝑜𝑙𝑒=𝑧0 = lim𝑧→𝑧0
(𝑧 − 𝑧0)𝑓(𝑧).
𝐶
𝑓(𝑧) = 2𝜋𝑖
𝑝𝑜𝑙𝑒𝑠
Re𝑠𝑖𝑑𝑢𝑒𝑠Final
Answer
Laurent Series
OR
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Calculating Residue’s
For a pole of Order ‘n’
Residue
𝐶
𝑓(𝑧) = 2𝜋𝑖
𝑝𝑜𝑙𝑒𝑠
Re𝑠𝑖𝑑𝑢𝑒𝑠Final
Answer
𝑓(𝑧) =)𝑓0(𝑧
𝑧 − 𝑧0𝑛=
𝑎𝑛𝑧 − 𝑧0
𝑛+. . . . . . . .
𝑎−1𝑧 − 𝑧0
+ 𝑎0 + 𝑎1(𝑧 − 𝑧0) + 𝑎1 𝑧 − 𝑧02+. . . .
Re𝑠𝑖𝑑𝑢𝑒|𝑃𝑜𝑙𝑒=𝑧0 =1
(𝑛 − 1)!lim𝑧→𝑧0
ቇ𝑑𝑛−1
𝑑𝑧𝑛−1𝑧 − 𝑧0
𝑛𝑓(𝑧
Laurent Series
OR
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Taking contour only in the upper half plane
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Proof of Cauchy’s Residue theorem by
Laurent Series
𝑓(𝑧) =)𝑓0(𝑧
𝑧 − 𝑧0𝑛=
𝑎𝑛𝑧 − 𝑧0
𝑛+. . . . . . . .
𝑎−1𝑧 − 𝑧0
+ 𝑎0 + 𝑎1(𝑧 − 𝑧0) + 𝑎1 𝑧 − 𝑧02+. . . .Laurent Series
Nth Coefficient : 𝑎𝑛 =1
2𝜋𝑖
𝐶
)𝑓(𝑧
𝑧 − 𝑧0𝑛+1 𝑑𝑧
We are interested in
this
So we take n=-1,Thus 𝑧 − 𝑧0
0 = 1
𝑎−1 =1
2𝜋𝑖ර
𝐶
)𝑓(𝑧 𝑑𝑧
ර
𝐶
)𝑓(𝑧 𝑑𝑧 = (2𝜋𝑖)𝑎−1
Thus we are so interested in the
Residue: coefficient of the
power 𝑧 − 𝑧0
−1
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Must See Videos
• https://youtu.be/T647CGsuOVU
• (You think you know about Complex Numbers: Decide after watching the above video: max 2 per day)
Other Videos (For Contour Integration)
https://youtu.be/b5VUnapu-qs
https://youtu.be/_3p_E9jZOU8
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ReferencesMathematical Methods for
Physics and Engineering. K.F. Riley ,M.P. Hobson, S.J.
Bence, Cambridge University Press
Mathematical Methods for Physicists
A Comprehensive Guide,George B. Arfken, Hans J.
Weber and Frank E. Harris, Elsevier
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Thank You