Applications of Residue Theorem to Four Distinct Kinds of Definite Integrals

Abstract

Cauchy’s residue theorem gives a relative general form for complex integral along a simple closed contour. With the help of Cauchy’s residue theorem, appropriate closed contour can be chosen to calculate some abnormal definite integrals that might be very complicated and are difficult to solve by conventional methods. This paper focuses on four distinct kinds of definite integrals, including the integrals involving sine and cosine functions, polynomial functions, exponential functions, and logarithmic functions. The contour chosen are basically a sector of circle that involves one or several isolated singularities of the function. Then the residue at the isolated singularities of the function is calculated. The value of the residues is substituted in the formula that is deducted in the Cauchy’s residue theorem. Then the integral along the simple closed contour can be expressed in two parts, in which one is along the real axis while the other is along the circle. This study demonstrates that the Cauchy’s residue theorem is superior to the conventional real analysis methods.