Calculus of Residues and Applications to Contour Integration

Abstract

The inspiration behind this chapter is the desire to obtain possible values for the integrals ${\int _{C}} f(z)\, dz$, 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.

Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality

Richard Courant

Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country

David Hilbert