Presented By Osman Toufiq Ist Year Ist SEM RESIDUE THEOREM Presented By Osman Toufiq Ist Year Ist SEM

RESIDUE THEOREM Calculus of residues Suppose an analytic function f (z) has an isolated singularity at z0. Consider a contour integral enclosing z0 . z0 The coefficient a-1=Res f (z0) in the Laurent expansion is called the residue of f (z) at z = z0. If the contour encloses multiple isolated singularities, we have the residue theorem: z0 z1 Contour integral =2pi ×Sum of the residues at the enclosed singular points

Residue formula: To find a residue, we need to do the Laurent expansion and pick up the coefficient a-1. However, in many cases we have a useful residue formula:

Residue at infinity: Stereographic projection: Suppose f (z) has only isolated singularities, then its residue at infinity is defined as Another way to prove it is to use Cauchy’s integral theorem. The contour integral for a small loop in an analytic region is One other equivalent way to calculate the residue at infinity is By this definition a function may be analytic at infinity but still has a residue there.

C r=1 z+ z- C r=1

C r=1 z+ z-

C r=1 z+ z- z0

Question: How about Answer: We can go the lower half of the complex plane using a clockwise contour.