Applications of Cauchy's Residue Theorem in Complex Functions of Fractional Form

Wenshuo Li

2025

Abstract

This paper explores the applications of Cauchy's Residue Theorem in complex analysis, focusing on its utility in evaluating complex integrals around closed contours. The study begins with an introduction to the Laurent series and Taylor series, which are foundational for understanding the Residue Theorem. The Residue Theorem uses these series to find "residues", which is a fancy term for coefficients that capture the behavior at singularities. Summing these residues gives the integral’s value instantly. The paper then delves into the theorem's theoretical framework, illustrating its application through several examples, including functions with simple poles, higher-order poles, and removable singularities. For simple poles (basic singularities), calculating residues is straightforward. For harder cases (like higher-order poles), it needs the use of derivatives. The results demonstrate the theorem's effectiveness in simplifying complex integral calculations, particularly in cases involving trigonometric and rational functions. The research highlights the theorem's significance in both theoretical mathematics and practical applications, such as physics and engineering. The paper concludes with a discussion on the potential for further exploration and the implications of these findings for advanced mathematical studies.

Paper Citation

in Harvard Style

Li W. (2025). Applications of Cauchy's Residue Theorem in Complex Functions of Fractional Form. In Proceedings of the 2nd International Conference on Innovations in Applied Mathematics, Physics, and Astronomy - Volume 1: IAMPA; ISBN 978-989-758-774-0, SciTePress, pages 99-103. DOI: 10.5220/0013814700004708

in Bibtex Style

@conference{iampa25,
author={Wenshuo Li},
title={Applications of Cauchy's Residue Theorem in Complex Functions of Fractional Form},
booktitle={Proceedings of the 2nd International Conference on Innovations in Applied Mathematics, Physics, and Astronomy - Volume 1: IAMPA},
year={2025},
pages={99-103},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0013814700004708},
isbn={978-989-758-774-0},
}

in EndNote Style

TY - CONF

JO - Proceedings of the 2nd International Conference on Innovations in Applied Mathematics, Physics, and Astronomy - Volume 1: IAMPA
TI - Applications of Cauchy's Residue Theorem in Complex Functions of Fractional Form
SN - 978-989-758-774-0
AU - Li W.
PY - 2025
SP - 99
EP - 103
DO - 10.5220/0013814700004708
PB - SciTePress