Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.12 RESIDUE THEOREM

Suppose the functionf(z) has a pole of ordermat the pointz=z 0 ,andso

can be written as a Laurent series aboutz 0 of the form

f(z)=

∑∞

n=−m

an(z−z 0 )n. (24.59)

Now consider the integralIoff(z) around a closed contourCthat encloses

z=z 0 , but no other singular points. Using Cauchy’s theorem, this integral has

the same value as the integral around a circleγof radiusρcentred onz=z 0 ,

sincef(z) is analytic in the region betweenCandγ. On the circle we have

z=z 0 +ρexpiθ(anddz=iρexpiθ dθ), and so

I=

∮

γ

f(z)dz

=

∑∞

n=−m

an

∮

(z−z 0 )ndz

=

∑∞

n=−m

an

∫ 2 π

0

iρn+1exp[i(n+1)θ]dθ.

For every term in the series withn=−1, we have

∫ 2 π

0

iρn+1exp[i(n+1)θ]dθ=

[

iρn+1exp[i(n+1)θ]

i(n+1)

] 2 π

0

=0,

but for then=−1termweobtain
∫ 2 π

0

idθ=2πi.

Therefore only the term in (z−z 0 )−^1 contributes to the value of the integral

aroundγ(and thereforeC), andItakes the value

I=

∮

C

f(z)dz=2πia− 1. (24.60)

Thus the integral around any closed contour containing a single pole of general

orderm(or, by extension, an essential singularity) is equal to 2πitimes the residue

off(z)atz=z 0.

If we extend the above argument to the case wheref(z) is continuous within

and on a closed contourCand analytic, except for a finite number of poles,

withinC, then we arrive at theresidue theorem
∮

C

f(z)dz=2πi

∑

j

Rj, (24.61)

where

∑

jRjis the sum of the residues off(z) at its poles withinC.

The method of proof is indicated by figure 24.13, in which (a) shows the original

contourCreferred to in (24.61) and (b) shows a contourC′giving the same value