Page 107 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 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 z is an isolated singular point of f . Then there is a Laurent series
0


f(z)   c (z z ) j
0
j
j – 
valid for 0 < |z – z | < R, for some positive R. The coefficient c of (z – z ) is called the residue
–1
0
–1
0
of f at z , and is frequently written
0
Res f.
z 0 z
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 – z | < R and which contains z inside, then
0
0
1
c   f(z)dz.
 1 2 i C

This provides the key to evaluating many complex integrals.
Example:
We shall evaluate the integral

 f 1/z dz
C
where C is the circle |z| = 1 with the usual positive orientation. Observe that the integrand has
an isolated singularity at z = 0. We know then that the value of the integral is simply 2i times
the residue of e at 0. Let’s find the Laurent series about 0. We already know that
1/z
 1
z
e   z j
j 0 z!

for all z. Thus,
 1 1 1 1
1
e 1/z   z     ...
j 
j 0 j! z 2! z 2

The residue c = 1, and so the value of the integral is simply 2i.
–1
Now suppose, we have a function f which is analytic everywhere except for isolated singularities,
and let C be a simple closed curve (positively oriented) on which f is analytic. Then there will be
only a finite number of singularities of f inside C (why?). Call them z , z , ..., z . For each
1
n
2
k = 1, 2, ..., n, let C be a positively oriented circle centered at z and with radius small enough to
k
k
insure that it is inside C and has no other singular points inside it.

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