Calculating Specific Integrals by Using Residue Theorem

Abstract

The aim of this paper is to solve the some specific integrals such as , , , and . The traditional method is that people normally decompose this polynomial into several partial fractions first. This process involves adding it all up, expanding brackets, and doing matrices computation, which takes too many steps of calculation. The partial fraction part requires using Euler’s formula and large amounts of expanding brackets to prove that the multiplication of those partial denominators is equal to the denominator given in the original equation. Once making one little mistake, the next process will be all nonsense. Therefore, introducing complex analysis and residue theorem can result in considerably fewer calculation steps than the traditional method does in order to calculate integral, as opposed to completing hundreds of steps of partial fraction decomposition and substitution. To solve the first integral: , we can rewrite Cauchy’s residue theorem in a new form, by considering residue at infinity , .Then, we can get the coefficient of  and hence compute the integral. For the second and third integral: , and , we exploit the reciprocal function of Taylor polynomials, writing  and  in a new form to get the coefficient of  Then we can compute the integrals by using the residue theorem.