The Application of Cauchy Integral Formula to Functions

Authors

  • Zijun Liang

DOI:

https://doi.org/10.61173/1m7bha73

Keywords:

Cauchy integral formula, Analytic function, Singular points

Abstract

The Cauchy integral formula is a fundamental and critical formula in the complex function, and it has a variety of applications in many aspects such as physics, engineering, and mathematics. This article focuses on the applications in the functions and two main theorems in the Cauchy integral formula. It explores the formula and provides corresponding proofs of the formula. To this end, the aim of the article is to use the Cauchy integral formula to cope with some difficult and complex integrals and functions. As a result, this article denotes to use the formula to simplify the complex and difficult problems and provides a practical and fast method. Hence, the article lists some examples to demonstrate the functions’ use of the Cauchy integral formula. At last, the article may serve as a reference for relevant research and survey in the field of the complex functions and provide insights into the Cauchy integral formula’s uses.

References

[1] V. S. Shpakivskyi; T. S. Kuzmenko. Integral theorems for the quaternionic G-monogenic mappings. Analele Universitatii “Ovidius” Constanta - Seria Matematica. 2016, 24(2): 271-281.

[2] Zhou, CM., Wu, L. The difference between the Cauchy Integral Formula and the Residue Theorem in calculating line integrals. Journal of Ningxia Normal University. 2018, 39(10): 94-97.

[3] Bell Steven R. The Cauchy Integral Formula, Quadrature Domains, and Riemann Mapping Theorems. Computational Methods and Function Theory. 2018, 18(4): 661-676. Dean&Francis ISSN 2959-6157

[4] Chen, YP., A Note on the Higher-Order Derivative Formulas of Cauchy Integral. College Mathematics, 2021, 37(6): 21-26.

[5] Wang, XP., Residue theorem and integral of complex variable function. Modern Women (late). 2014, (01), 247-248.

[6] James Ward Brown; Ruel V. Churchill. Complex Variables and Applications. McGraw-Hill Education. New York, (2014).

[7] Harald Hanche-Olsen. On Gour sat’s Proof of Cauchy’s Integral Theorem. The American Mathematical Monthly. 2008, 115(7), 648-652.

[8] Zhao, YN., Wang ZH. The application of Cauchy integral theorem and Cauchy integral formula to real functions. Examination weekly, 2011.

Downloads

Published

2025-06-26

Issue

Vol. 3 No. 3 (2025): Volume 3, Issue 3

Section

Articles

License

Copyright (c) 2025 by the authors.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.