Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS OF A COMPLEX VARIABLE
Solution:
We ®rst apply the method of partial fractions to f …z† and obtain
f …z† ˆ 3
z ‡ 1 ‡ 1 z ‡ 2
z 2 :
Now the center of the given annulus is z ˆ1, so the series we are seeking must
be one involving powers of z ‡ 1. This means that we have to modify the second
and third terms in the partial fraction representation of f …z†:
f …z† ˆ 3
z ‡ 1 ‡ 1
…z ‡ 1†1 ‡ 2
…z ‡ 1†3 ;
but the series for ‰…z ‡ 1†3Š 1 converges only where jz ‡ 1j > 3, whereas we
require an expansion valid for jz ‡ 1j < 3. Hence we rewrite the third term in
the other order:
f …z† ˆ 3
z ‡ 1 ‡ 1
…z ‡ 1†1 ‡ 2
3 ‡…z ‡ 1†
ˆ3…z ‡ 1† 1 ‡‰…z ‡ 1†1Š 1 ‡ 2‰3 ‡…z ‡ 1†Š 1
ˆ‡…z ‡ 1† 2 2…z ‡ 1† 1 2 3 2 …z ‡ 1†
9
2
27 …z ‡ 1†2 ; 1 < jz ‡ 1j < 3:
Example 6.24
Given the following two functions:
…a† e 3z …z ‡ 1† 3 ; …b† …z ‡ 2† sin 1
z ‡ 2 ;
®nd Laurent series about the singularity for each of the functions, name the
singularity, and give the region of convergence.
Solution: (a) z ˆ1 is a triple pole (pole of order 3). Let z ‡ 1 ˆ u, then
z ˆ u 1 and
!
e 3z
…z ‡ 1† 3 ˆ e3…u1†
u 3 ˆ e 3 e 3u
u 3 ˆ e3
u 3 1 ‡ 3u ‡ …3u†2 ‡ …3u†3 ‡ …3u†4 ‡
2! 3! 4!
ˆ e 3 1
…z ‡ 1† 3 ‡ 3
…z ‡ 1† 2 ‡ 9
2…z ‡ 1† ‡ 9 2
27…z ‡ 1†
‡ ‡
8
The series converges for all values of z 6ˆ 1.
(b) z ˆ2 is an essential singularity. Let z ‡ 2 ˆ u, then z ˆ u 2, and
278
!
: