Complex analysis and residue theorem

Nai-Jun Wang

doi:10.1117/12.2672666

28 March 2023 Complex analysis and residue theorem

Nai-Jun Wang

Proceedings Volume 12597, Second International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2022); 125970B (2023) https://doi.org/10.1117/12.2672666
Event: Second International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2022), 2022, Nanjing, China

Abstract

Complex analysis is a mathematical theory of complex functions, especially meromorphic functions and complex analytic functions, and it generalizes analytical techniques from real variables to complex variables. Complex numbers originally arose from the fact that universal solutions could exist for algebraic equations. They take the form x + iy, x and 𝑦 are real numbers. X is called the real section of the complex number, y is the imaginary section, and i is the square root of -1, the imaginary unit. Because complex numbers have two separated components, 𝑥 and 𝑦, they are especially useful when two variables must be processed simultaneously. For example, it has proved particularly valuable in fluid dynamics, where pressure and velocity vary from place to place. In the mid-19th century, mathematicians gave complex numbers a geometric interpretation to make them more acceptable. In this paper, we mainly focus on the applications of Cauchy Residue Theorem by using a specific example. And the method of calculating residues, which is called Fourier Matrix, are also included. As a result, all the three different integrals of a complex function can be correctly evaluated by using Cauchy Residue Theorem.

Citation Download Citation

Nai-Jun Wang "Complex analysis and residue theorem", Proc. SPIE 12597, Second International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2022), 125970B (28 March 2023); https://doi.org/10.1117/12.2672666

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