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.