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Integral Theorems

Table of Contents

1. Cauchy's Integral Theroem

If f is holomorphic on a simply connected domain D, and γ is a closed curve, then:

γf(z)dz=0

2. Cauchy's Integral Formula

Let:

  1. D be the closed disk D={z:|zz0|r} centered at z0
  2. f:DC be a holomorphic function
  3. γ be the counterclockwise circle forming the boundary of D

Then:

f(a)=12πiγf(z)zadz

Holds for all a in the interior of D.

3. Cauchy's Residue Theorem

Let:

  1. anU be some finite set of points
  2. D=Uan be a simply connected domain
  3. f:DC a holomorphic function
  4. γ a closed contour in D

Then:

γf(z)dz=2πiRes(f,ak)

It is worth noting that CRT is a generalisation of both of the theorems above, the first follows by noting an is empty. For the second:

γf(z)zadz=2πi Res(f(z)za,a)=2πi f(a)

Author: root

Created: 2025-02-15 Sat 15:26

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