文档介绍：Chapter Ten
Poles, Residues, and All That
. Residues. A point z0 is a singular point of a function f if f not analytic at z0, but is analytic
at some point of each neighborhood of z0. A singular point z0 of f is said to be isolated if there is a
neighborhood of z which contains no singular points of f save z . In other words, f is analytic on
< 0? < P 0
some region 0 |z z0 | .
Examples
The function f given by
fÝzÞ = 1
zÝz2 + 4Þ
?
has isolated singular points at z = 0, z = 2i, and z = 2i.
Every point on the negative real axis and the origin is a singular point of Log z , but there are no
isolated singular points.
Suppose now that z0 is an isolated singular point of f . Then there is a Laurent series
K
Ý Þ = > Ý ? Þj
f z cj z z0
?K
j=
?
< ? < ? Ý ? Þ 1
valid for 0 |z z0 | R, for some positive R. The coefficient c 1 of z z0 is called the
residue of f at z0, and is frequently written
Res f.
=
z z0
Now, why do we care enough about c? to give it a special name? Well, observe that if C is any
1 < ? <
positively oriented simple closed curve in 0 |z z0 | R and which contains z0 inside, then
X
c = 1 fÝzÞdz.
j 2^i
C
This provides the key to evaluating plex integrals.
Example
We shall evaluate the integral
X
e1/zdz
C
=
where C is the circle |z| 1 with the usual positive orientation. Observe that the integra