Complex Analysis

7

Applications of Residues

7.1 INTRODUCTION

In the last chapter, we have studied the theory of residues. Now, we will study some important applications of the theory of residues. The applications include evaluation of certain types of real definite and improper integrals. These integrals can be evaluated with the help of contour integration and residue theorem.

7.2 DEFINITE INTEGRALS INVOLVING SINES AND COSINES

Let us consider the definite integrals of the form

where f(sin θ, cos θ) is a real rational function of sin θ and cos θ. As θ varies from 0 to 2π or ‒ π to π, it is considered as the argument of a point z on a positively oriented unit ...