Theoretical and Natural Science

- The Open Access Proceedings Series for Conferences

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

Series Vol. 5 , 25 May 2023

Open Access | Article

### Complex Analysis and Residue Theorem

* Author to whom correspondence should be addressed.

Theoretical and Natural Science, Vol. 5, 95-99
Citation Zixuan Xia. Complex Analysis and Residue Theorem. TNS (2023) Vol. 5: 95-99. DOI: 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.

### References

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Volume Title
Proceedings of the 2nd International Conference on Computing Innovation and Applied Physics (CONF-CIAP 2023)
ISBN (Print)
978-1-915371-53-9
ISBN (Online)
978-1-915371-54-6
Published Date
25 May 2023
Series
Theoretical and Natural Science
ISSN (Print)
2753-8818
ISSN (Online)
2753-8826
DOI
10.54254/2753-8818/5/20230307