Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX VARIABLES

x

y

R

z 0

C 1

C 2

Figure 24.12 The region of convergenceRfor a Laurent series off(z) about

a pointz=z 0 wheref(z) has a singularity.

classify the nature of that point. Iff(z) is actually analytic atz=z 0 ,thenin

(24.55) allanforn<0 must be zero. It may happen that not only are allan

zero forn<0 buta 0 ,a 1 ,...,am− 1 are all zero as well. In this case, the first

non-vanishing term in (24.55) isam(z−z 0 )m, withm>0, andf(z)isthensaidto

have azero of ordermatz=z 0.

Iff(z) is not analytic atz=z 0 , then two cases arise, as discussed above (pis

here taken as positive):

(i) it is possible to find an integerpsuch thata−p= 0 buta−p−k= 0 for all

integersk>0;

(ii) it is not possible to find such a lowest value of−p.

In case (i),f(z) is of the form (24.54) and is described as having apole of order

patz=z 0 ; the value ofa− 1 (nota−p) is called theresidueoff(z) at the pole

z=z 0 , and will play an important part in later applications.

For case (ii), in which the negatively decreasing powers ofz−z 0 do not

terminate,f(z) is said to have anessential singularity. These definitions should be

compared with those given in section 24.6.

Find the Laurent series of

f(z)=

1

z(z−2)^3

about the singularitiesz=0andz=2(separately). Hence verify thatz=0is a pole of

order 1 andz=2is a pole of order 3 , and find the residue off(z)at each pole.

To obtain the Laurent series aboutz= 0, we make the factor in parentheses in the