Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.11 TAYLOR AND LAURENT SERIES

of orderpatz=z 0 but is analytic at every other point inside and onC.Then

the functiong(z)=(z−z 0 )pf(z) is analytic atz=z 0 , and so may be expanded

as a Taylor series aboutz=z 0 :

g(z)=

∑∞

n=0

bn(z−z 0 )n. (24.53)

Thus, for allzinsideC,f(z) will have a power series representation of the form

f(z)=

a−p

(z−z 0 )p

+···+

a− 1

z−z 0

+a 0 +a 1 (z−z 0 )+a 2 (z−z 0 )^2 +···,

(24.54)

witha−p= 0. Such a series, which is an extension of the Taylor expansion, is

called aLaurent series. By comparing the coefficients in (24.53) and (24.54), we

see thatan=bn+p. Now, the coefficientsbnin the Taylor expansion ofg(z)are

seen from (24.52) to be given by

bn=

g(n)(z 0 )

n!

=

1

2 πi

∮

g(z)

(z−z 0 )n+1

dz,

and so for the coefficientsanin (24.54) we have

an=

1

2 πi

∮

g(z)

(z−z 0 )n+1+p

dz=

1

2 πi

∮

f(z)

(z−z 0 )n+1

dz,

an expression that is valid for both positive and negativen.

The terms in the Laurent series withn≥0 are collectively called theanalytic

part, whilst the remainder of the series, consisting of terms in inverse powers of

z−z 0 , is called theprincipal part. Depending on the nature of the pointz=z 0 ,

the principal part may contain an infinite number of terms, so that

f(z)=

∑+∞

n=−∞

an(z−z 0 )n. (24.55)

In this case we would expect the principal part to converge only for|(z−z 0 )−^1 |

less than some constant, i.e.outsidesome circle centred onz 0. However, the

analytic part will convergeinsidesome (different) circle also centred onz 0 .Ifthe

latter circle has the greater radius then the Laurent series will converge in the

regionRbetweenthe two circles (see figure 24.12); otherwise it does not converge

at all.

In fact, it may be shown that any functionf(z) that is analytic in a region

Rbetween two such circlesC 1 andC 2 centred onz=z 0 can be expressed as

a Laurent series aboutz 0 that converges inR. We note that, depending on the

nature of the pointz=z 0 , the inner circle may be a point (when the principal

part contains only a finite number of terms) and the outer circle may have an

infinite radius.

We may use the Laurent series of a functionf(z) about any pointz=z 0 to