Complex Analysis and Residue Theorem

Abstract

Mathematics is knitted into our lives by laying the fundamental base of many things. But people still think it is unnecessary to study math deeply due to its complexity. A lot of the great work from mathematicians is not noticed by other fields. Among all the essential mathematical methods and theorems, the Residue Theorem is one of the most significant ones. Therefore, this article will interpret the Residue Theorem step by step with its related theorems. Before jumping into the Residue Theorem, the article will introduce a series of mathematical methods, including the Triangle Inequality, the Euler's Formula, the Analytic function, the Taylor Series, and the Laurent Series. The applications and graphs of these mathematical methods will help to understand them further. Consequently, this article will demonstrate the most crucial part of this article — the Residue Theorem by its definition, formula, and example. Last but not least, this article will finish with how the Residue Theorem is used in our wind power generation system to express further how math correlates to our daily life. The result of this article shows that the Residue Theorem can be used to help explain other theorems and to extend new mathematical theories. It can be used not only in mathematics but also in a lot of fields, such as computer science, physics, and engineering. This article hopes a more extensive population can acknowledge the contribution of math to our world through the Residue Theorem.