Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.11 Taylor and Laurent series

Further, it may be proved by induction that thenth derivative off(z) is also

given by a Cauchy integral,

f(n)(z 0 )=

n!

2 πi

∮

C

f(z)dz

(z−z 0 )n+1

. (24.48)

Thus, if the value of the analytic function is known onCthen not only may the

value of the function at any interior point be calculated, but also the values of

allits derivatives.

The observant reader will notice that (24.48) may also be obtained by the

formal device of differentiating under the integral sign with respect toz 0 in

Cauchy’s integral formula (24.46):

f(n)(z 0 )=

1

2 πi

∮

C

∂n

∂zn 0

[

f(z)

(z−z 0 )

]

dz

=

n!

2 πi

∮

C

f(z)dz

(z−z 0 )n+1

.

Suppose thatf(z)is analytic inside and on a circleCof radiusRcentred on the point

z=z 0 .If|f(z)|≤Mon the circle, whereMis some constant, show that

|f(n)(z 0 )|≤

Mn!

Rn

. (24.49)

From (24.48) we have

|f(n)(z 0 )|=

n!

2 π

∣

∣

∣∣

∮

C

f(z)dz

(z−z 0 )n+1

∣

∣

∣∣,

and on using (24.39) this becomes

|f(n)(z 0 )|≤

n!

2 π

M

Rn+1

2 πR=

Mn!

Rn

.

This result is known asCauchy’s inequality.

We may use Cauchy’s inequality to proveLiouville’s theorem, which states that

iff(z) is analytic and bounded for allzthenfis a constant. Settingn=1in

(24.49) and lettingR→∞, we find|f′(z 0 )|=0andhencef′(z 0 ) = 0. Sincef(z)is

analytic for allz,wemaytakez 0 as any point in thez-plane and thusf′(z)=0

for allz; this impliesf(z) = constant. Liouville’s theorem may be used in turn to

prove thefundamental theorem of algebra(see exercise 24.9).

24.11 Taylor and Laurent series

Following on from (24.48), we may establishTaylor’s theoremforfunctionsofa

complex variable. Iff(z) is analytic inside and on a circleCof radiusRcentred

on the pointz=z 0 ,andzis a point insideC,then

f(z)=

∑∞

n=0

an(z−z 0 )n, (24.50)