Abstract
Calculating definite integrals in complex functions requires the Cauchy's residue theorem, which is a key concept in the complex variables. It is based on several ideas, including the isolated singular points theory, the Laurent theorem, and the Cauchy integral theorem. Nevertheless, it is highly challenging to calculate closed curves using the conventional integrals method since numerous formulas must be substituted. The singular points must all be isolated in the case of a holomorphic function that belongs to an enclosing contour and is positively oriented, at which time the contour integral can be used to resolve the issue. Additionally, the problem can be solved by using visual aids to help people understand them. By doing so, the notion of solving the problem will be made clearer and some calculating steps will be cut short. This study exhibits the residue theorem's potent efficiency in computing improper integrals and other difficult problems.
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