Overview
You now have the necessary machinery to see some amazing applications of
the tools we developed in the last few chapters. You will learn how Laurent
expansions can give useful i nformation concerning seemingly unrealated proper-
ties 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-
literally-as the computation of residues. We begin with a theorem relating
residues to the evaluation of complex integrals.
8.1 The Residue Theorem
The Cauchy integral formulas given in Section 6.5 are useful in evaluating contour
integrals over a simple closed contour C where the integrand has the form ~( z-zo / z J
and f is an analytic function. In this case, the singularity of the integrand is
at worst a pole of order k at zo. We begin this section by extending this result
to integrals that have a finite number of isolated singularities inside the contour
C. This new method can be used in cases where the integrand has an es.sential
singularity at ZQ and is an important extension of the previous method.
Definition 8.1: Residue
Let f have a nonremovable isolated singularity at the point zo. Then f has the
Laurent series represer1tation for all z in some disk Dn (zo) given by f (z) =
00
:L a., (z - zo)". The coefficient a_ 1 is called the residue of fat zo. We use
n=- oo
the not ation
Res[/,zo] = a-1·
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