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Cauchy’s residue theorem is a significant theorem in complex number analysis. Motivated by the residue theorem built by Cauchy, numerous researchers have studied and tried to apply this theorem in different academic fields. In this paper, we are going to generalize Cauchy’s residue theorem in multiple fields. Firstly, we will review the residue theorem and some theorems developed from the residue theorem. Then a definite integral of function will be calculated as an example to show the application of residue theorem in real number calculation. Next, we discuss the theorems of logarithmic residue which is discovered by utilizing residue theorem, and a short proof will be used to illustrate the connection between different logarithmic residue theorems. Finally, based on Jacobi’s residue theorem, we will have a brief synopsis of the application of residue theorem in combinatorics. By these generalizations, we find that the residue theorem can be widely applied in different academic aspects.
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