Complex Analysis and Residue Theorem

Zixuan Xia

doi:10.54254/2753-8818/5/20230307

Research Article

Open access

Published on 25 May 2023

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Xia,Z. (2023). Complex Analysis and Residue Theorem. Theoretical and Natural Science,5,95-99.

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Complex Analysis and Residue Theorem

Zixuan Xia *^,1,

¹ Minhang Crosspoint High School, No.980, Jiangchuan East Road, Minhang District., Shanghai, China

* Author to whom correspondence should be addressed.

https://doi.org/10.54254/2753-8818/5/20230307

Abstract

The study of the properties of analytical functions is described as complex analysis. The residue theorem is an important conclusion in complex analysis. This paper introduces the origin of imaginary numbers from Cardano Formula defines the conversion of complex number formats from the Euler Theorem, and proves the intermediate theorem Cauchy Integral Formula before reaching our final conclusion and the goal of the paper, Residue Theorem. The name of the theorem comes from the concept of residue, which is defined using a function’s Laurent series. We could then derive the Residue Theorem from the Cauchy Integral Theorem, also called the Cauchy-Goursat Theorem. We will be able to formalize our prior, ad hoc method of computing integrals over contours encompassing singularities. Additionally, it is a theorem that may be applied to zero-pole qualities and curved integral properties. The Residue Theorem is the basis of many essential mathematical facts revolving around line integrals, particularly in solving ODEs and PDEs and describing physics models.

Keywords

residue theorem, complex analysis, residue.

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Cite this article

Xia,Z. (2023). Complex Analysis and Residue Theorem. Theoretical and Natural Science,5,95-99.

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About volume

Volume title: Proceedings of the 2nd International Conference on Computing Innovation and Applied Physics (CONF-CIAP 2023)

Conference website: https://www.confciap.org/

ISBN：978-1-915371-53-9(Print) / 978-1-915371-54-6(Online)

Conference date: 25 March 2023

Editor：Marwan Omar, Roman Bauer

Series: Theoretical and Natural Science

Volume number: Vol.5

ISSN：2753-8818(Print) / 2753-8826(Online)

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