Cauchy’s Integral Formula and Its Applications in Calculating Integrals

Abstract

Cauchy's Integral Theorem is a crucial mathematical theorem that describes the behavior of a complex function on a complex plane. This paper reviews the proof of Cauchy's integral theorem and formula, and the deduction of Cauchy's integral like Cauchy's integral formula with the point in the numerator is on the contour integral. This paper also explores Cauchy's integral when the function contains a non-analytic point inside the contour integral. The way is splitting the entire integral into the sum of several contour integrals equivalently, putting the function f(z) in the numerator into Laurent Series, then simplifying each contour integral using different methods including the derivative of Cauchy’s Integral. The result is the contour integral around point  minus part of the Laurent Series of f(x) in the numerator. This paper shows a path to deal with Cauchy's integral when the function inside the contour integral is not holomorphic in the region.