Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX VARIABLES

which is an alternating series whose terms decrease in magnitude and which

therefore converges.

The ratio test discussed in subsection 4.3.2 may also be employed to investi-

gate the absolute convergence of a complex power series. A series is absolutely

convergent if

lim

n→∞

|an+1||z|n+1

|an||z|n

= lim

n→∞

|an+1||z|

|an|

< 1 (24.14)

and hence the radius of convergenceRof the series is given by

1

R

= lim

n→∞

|an+1|

|an|

For instance, in case (i) of the previous example, we have

1

R

= lim

n→∞

n!

(n+1)!

= lim

n→∞

1

n+1

=0.

Thus the series is absolutely convergent for all (finite)z, confirming the previous

result.

Before turning to particular power series, we conclude this section by stating

the important result§thatthe power series

∑∞

0 anz

nhas a sum that is an analytic

function ofzinside its circle of convergence.

∑As a corollary to the above theorem, it may further be shown that iff(z)=
anznthen, inside the circle of convergence of the series,

f′(z)=

∑∞

n=0

nanzn−^1.

Repeated application of this result demonstrates that any power series can be

differentiated any number of times inside its circle of convergence.

24.4 Some elementary functions

In the example at the end of the previous section it was shown that the function

expzdefinedby

expz=

∑∞

n=0

zn

n!

(24.15)

is convergent for allz of finite modulus and is thus, by the discussion of

the previous section, an analytic function over the wholez-plane.¶ Like its

§For a proof see, for example, K. F. Riley,Mathematical Methods for the Physical Sciences(Cam-

bridge: Cambridge University Press, 1974), p. 446.

¶Functions that are analytic in thewholez-plane are usually calledintegralorentirefunctions.