Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS OF A COMPLEX VARIABLE
Ratio test
Given the series f 1 …z†‡f 2 …z†‡f 3 …z†‡‡f n …z†‡, the series converges absolutely
if
f
0 < jr…z†
j ˆ lim n‡1 …z†
n!1 f n …z† < 1
…6:30†
and diverges if jr…z†j > 1. When jr…z†j ˆ 1, the ratio test provides no information
about the convergence or divergence of the series.
Example 6.19
Consider the complex series
X
S n ˆ X1
n
nˆ0
… 2 n ‡ ie n † ˆ X1
nˆ0
2 n ‡ i X1
nˆ0
e n :
The ratio tests on the real and imaginary parts show that both converge:
2 …n‡1†
lim
n!1
2 n
ˆ 1 , which is positive and less than 1;
2 e …n‡1†
lim
n!1
e n
ˆ 1 , which is also positive and less than 1.
e
One can prove that the full series converges to
X 1
nˆ1
S n ˆ
1
1 1=2 ‡ i 1
1 e 1:
Uniform convergence and the Weierstrass M-test
To establish conditions, under which series can legitimately be integrated or
di€erentiated term by term, the concept of uniform convergence is required:
A series of functions is said to converge uniformly to the function
S(z) in a region R, either open or closed, if corresponding to an
arbitrary " N:
One of the tests for uniform convergence is the Weierstrass M-test (a sucient
test).
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