Higher-order of poles in dealing with complex plane analysis

Jiayi Guo; Meixin Ma; Jiajie Yao

doi:10.1117/12.2628080

22 April 2022 Higher-order of poles in dealing with complex plane analysis

Jiayi Guo, Meixin Ma, Jiajie Yao

Proceedings Volume 12163, International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2021); 121632Q (2022) https://doi.org/10.1117/12.2628080
Event: International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2021), 2021, Nanjing, China

Abstract

It's always difficult to solve when exploring the integral along contours in the complex plane if we meet several special cases. In this paper, we introduce a way to solve the integral of the function, which has a higher-order of poles. The residue theorem and its limit formula are effective methods to apply. Contrasting to the simple pole case, the higher-order case needs to concentrate more on the contour graph. Understanding how the graph comes is really beneficial for the deduction. From reading the figure of the contour, we can list an equation. The left-hand side will be the calculation of the residue theorem, which is the graphical method. The right-hand side will be the summation of integrals, including four pieces, which is the algebraic method. By distinguishing its real part and complex part, we can finish the proof. After finishing the proof, it’s easier for readers to solve higher-order problems that do not just limit to second-order problems.

Citation Download Citation

Jiayi Guo, Meixin Ma, and Jiajie Yao "Higher-order of poles in dealing with complex plane analysis", Proc. SPIE 12163, International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2021), 121632Q (22 April 2022); https://doi.org/10.1117/12.2628080

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