Practice Problems on Cauchy Theorem
Q1: Prove Cauchy’s Integral Theorem for a function f(z) that is analytic within a closed contour ( C ) and its interior.
Q2: Evaluate the contour integral ([Tex]\oint_C \frac{\sin z}{z} \, dz [/Tex] ) where ( C ) is the unit circle centered at the origin.
Q3: Apply Cauchy’s Residue Theorem to compute the integral ( [Tex]\oint_C \frac{e^z}{z^2 + 1} \, dz [/Tex]) where ( C ) is the contour ( |z| = 2 ).
Q4: Find the residue of ( [Tex]f(z) = \frac{e^z}{z^2 – 4} [/Tex]) at each singularity, and use it to compute the integral ( ∮Cf(z), dz ) along the contour ( C ) where ( C ) is the circle ( |z| = 3 ).
Q5: Using Cauchy’s Integral Formula, determine the value of [Tex]\oint_C \frac{e^z}{z} \, dz[/Tex] where ( C ) is the contour given by the line segment from z = 1 to z = 2i, followed by the line segment from z = 2i to z = -1.
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