The inspiration behind this chapter is the desire to obtain possible values for the integrals

∫ C f(z) dz, where f is analytic inside the closed curve C and on

C, except for a inside C. If f has a removable singularity at a, then it is clear that the integral will be zero. If z = a is a pole or an essential singularity, then the answer is not always zero, but can be found with little difficulty. In this chapter, we show the very surprising fact that Cauchy’s residue theorem yields a very elegant and simple method for evaluation of such integrals.