Cauchy’s Integral Theorem
Cauchy’s Integral Theorem is a fundamental concept in complex analysis. It states that if a function is analytic (meaning it has derivatives) within a closed contour (a loop) and its interior, then the integral of that function around the contour is zero. This theorem is widely used in various branches of mathematics and physics to solve problems involving complex functions and integrals.
Cauchy Theorem Statement
Cauchy’s Integral Theorem states that
If f(z) is analytic throughout a simply connected region containing a closed contour C, then the integral of f(z) around C is equal to zero.
Simply Connected Region
A simply connected region is a region where any loop can be continuously contracted to a point without leaving the region, and any two points can be connected by a path within the region.
A region D in the complex plane (or in any topological space) is simply connected if it has two properties:
- Path-Connected: Any two points in D can be connected by a continuous path that lies entirely within D.
- No Holes: Any closed loop (a continuous function from a circle into the region) in D can be continuously shrunk to a single point within D. This means there are no “holes” in the region.
Mathematical Formulation for Cauchy Theorem
Mathematical formulation of Cauchy’s Integral Theorem is expressed as follows:
∮(f(z) dz) = 0
Where:
- ∮ denotes the contour integral around a closed contour.
- f(z) is an analytic function defined within and on the contour.
- dz represents an infinitesimal displacement along the contour.
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