You have now the necessary machinery to see some amazing applications of
the tools that we had developed in the last few chapters. You will learn how Laurent
expansions can give useful information concerning seemingly unrelated properties of
complex functions. You will also learn how the ideas of complex analysis make the
solution of very complicated integrals of real-valued functions as easy – valued
functions as easy – literally – as the computation of complex integrals. The topic of our
discussion in this chapter is one of the most important and most often used tools that
applied scientists extract from the theory of complex variables. Residue theory is
employed in a wide variety of applications ranging from the evaluation of real integrals
to the stability of linear systems to image evaluation in photographic science. In this
chapter, we develop and illustrate some of the fundamental techniques employed in
complex integration by the use of residue theory.
Keywords: Application of residue theory to real definite integrals, improper
integrals, the residue theorem.
