Cauchy’s Residue Theorem
Cauchy’s Residue Theorem is a fundamental result in complex analysis that provides a powerful method for computing contour integrals of functions with singularities. It states that if f(z) is analytic inside and on a simple closed contour C, except at a finite number of isolated singular points z1,z2,…,zn, then the contour integral of f(z) around C is equal to 2πi times the sum of the residues of f(z) at the singular points inside C. Mathematically, it can be expressed as:
[Tex]\oint_C f(z) \, dz = 2\pi i \sum_{k=1}^n \text{Res}(f, z_k)[/Tex]
Where:
- ∮C denotes the contour integral around the closed contour C
- f(z) is a function analytic inside and on C, except at isolated singular points z1,z2,…,zn
- (f, zk) represents the residue of f(z) at the singular point zk
Cauchy Theorem
the Cauchy-GoursatCauchy’s Theorem states that if a function is analytic within a closed contour and its interior, then the integral of that function around the contour is zero. This fundamental result is central to complex analysis, providing insights into the behavior of analytic functions in complex domains.
In this article, we will learn about Cauchy’s Integral Theorem, Cauchy’s Integral Formula, It’s applications, Cauchy’s Residue Theorem and the Cauchy-Goursat Theorem.
Table of Content
- What is Cauchy’s Integral Theorem?
- Cauchy’s Integral Formula
- Applications of Cauchy Theorem
- Cauchy’s Residue Theorem
- Extension of Cauchy’s Theorem