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Complex analysis is the main branch of complex functions. It is widely utilized in many fields, including applied mathematics and physics, including algebra, combinatorial mathematics, nuclear engineering, aerospace engineering, and fluid mechanics. Thus, we need to dive into complex variables to take advantage of them in abundant fields. To explore complex variables in-depth, we learn and apply a lot of theorems. In this paper, particularly, we aim to evaluate a complicated integral in which f (z) is defined as (az3 + bz2 + cz + d)/(z4 - 1) with a = 10, b = 1 + i, c = -4, d = 1 - i. To solve this problem, we need to delve deeper into Cauchy-Goursat theorem to calculate those analytic points. However, it is often the case that we face some numbers which are not analytic, which means in this situation, Cauchy-Goursat theorem is not useful for us. Then, on these occasions, it seems necessary for us to use Cauchy’s residue theorem, for it is especially useful in terms of points that cannot be analyzed. Afterwards, we use plenty of mathematics strategies such as computation and combining and simplifying equations to secure the final results of the problem. Eventually, we need to put all our residues into the graphs.
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