Proof and Application of Cauchy’s Residue Theorem

Abstract

Complex analysis is a major subfield of mathematics that is concerned with investigating complex functions and their behaviors. The Cauchy’s residue theorem plays an important role in complex analysis. It is also the main focus for this paper. The residue theorem connects complex integrals of functions with their residues at singular points. It is widely used to find integrals of complex functions that may be insolvable otherwise. This paper first looks at the definition of Cauchy’s Residue Theorem. Next, it provides proof for the theorem from Laurent theorem and Cauchy-Goursat theorem. Finally, it looks at some applications of Cauchy’s Residue theorem, including the theorem’s use in evaluating integrals, in the proof of related theorems important to the field of complex analysis, and in its use in the formation of Fourier and Laplace transforms. Through these examples, the efficacy of the theorem, and the use of it in many areas, can be seen clearly.