Introduction

In this chapter we introduce the methods by which certain types of definite integral may be evaluated. Similar methods may be used to sum certain types of infinite series. The approach has many applications, and will be considered again in Chapter 16, in applications to Fourier transforms, and in Chapter 17, on Laplace transforms. We begin by establishing the Residue theorem, which relates a contour integral to the residues of the integrand at its various singularities. Then we explore how various types of real integral can be transformed into contour integrals, and then evaluated by an analysis of their singularities. Finally we take a brief look at the summation of series by residue methods.

Mathematica can play various roles in this part of the theory related to the evaluation of integrals by the calculus of residues. It can just be there to help with the algebra in calculating residues. You can use the functions Residue and NResidue to work out the residues directly. Finally you can use Integrate and NIntegrate to do a direct calculation of the answer. In this last case considerable care is required. The symbolic treatment of general integrals is an evolving (black) art and the results, mostly in the way they are displayed and the full details of conditions for the results to hold, will vary from version to version of the software. This matters particularly when the integrand contains parameters.