# The Residue Theorem: Applications in Complex Integrals

• Ruobing Niu

## Keywords:

Residue theorem; Complex analysis; Singularity; Infinite integral.

## Abstract

Complex function is a function that takes complex numbers as independent and dependent variables. The theory of complex functions originated in the eighteenth century, and in nineteenth century, the new branch of complex functions dominated mathematics. The theory about complex function is called complex analysis, which is very helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics as well as Physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. This paper is going to introduce an important theorem in complex analysis, and it is the residue theorem. The residue theorem can convert some integrals into integrals with complex variables, which can be solved by using Taylor series and it’s an easier method. With the explanation of those examples, the applications of using the theorem to solve functions are clearly shown. This paper is introducing several effective methods of how the residue theorem can be used in practical issues by solving integrals.

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29-03-2024

Articles

## How to Cite

Niu, R. (2024). The Residue Theorem: Applications in Complex Integrals. Highlights in Science, Engineering and Technology, 88, 458-463. https://doi.org/10.54097/3hb24375