Mathematical Methods for Physicists: A concise introduction - Site Map

SERIES REPRESENTATIONS OF ANALYTIC FUNCTIONS
If a sequence of positive constants fM n g exists such that
j f n …z†j M n for all positive integers n and for all values of z in
a given region R, and if the series
is convergent, then the series
converges uniformly in R.
M 1 ‡ M 2 ‡‡M n ‡
f 1 …z†‡f 2 …z†‡f 3 …z†‡‡f n …z†‡
As an illustrative example, we use it to test for uniform convergence of the
series
in the region jzj 1. Now
X 1
nˆ1
u n ˆ X1
nˆ1
z n
p
n
n ‡ 1
ju n jˆ p jzjn
n
1
n ‡ 1
if jzj 1. Calling M n ˆ 1=n 3=2 , we see that P M n converges, as it is a p series with
p ˆ 3=2. Hence by Wierstrass M-test the given series converges uniformly (and
absolutely) in the indicated region jzj 1.
n 3=2
Power series and Taylor series
Power series are one of the most important tools of complex analysis, as power
series with non-zero radii of convergence represent analytic functions. As an
example, the power series
S ˆ X1
a n z n
…6:31†
nˆ0
clearly de®nes an analytic function as long as the series converge. We will only be
interested in absolute convergence. Thus we have
a
lim n‡1 z n‡1
n!1
a n z n
< 1 or jj< z R ˆ lim ja n j
n!1 j j ;
where R is the radius of convergence since the series converges for all z lying
strictly inside a circle of radius R centered at the origin. Similarly, the series
S ˆ X1
nˆ0
a n …z z 0 † n
converges within a circle of radius R centered at z 0 .
269
a n‡1