Page 112 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 10: Residues and Singularities

10.3 Summary Notes

A point z is a singular point of a function f if f not analytic at z , but is analytic at some
 0 0
point of each neighborhood of z . A singular point z of f is said to be isolated if there is a
0
0
neighborhood of z which contains no singular points of f save z . In other words, f is
0
0
analytic on some region 0 < |z – z | < .
0
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 k = 1, 2, ..., n, let C be a positively oriented circle centered at z and with
1
2
n
k
k
radius small enough to insure that it is inside C and has no other singular points inside it.
Then,
 f(z)dz =  f(z)dz   f(z)dz ...   f(z)dz

C C 1 C2 C n
= 2 i Res f 2 iResf ... 2 iRes f     
z 1 z z 2 z z n z
n
= 2 i  Res f.
k 1 z k z

This is the celebrated Residue Theorem. It says that the integral of f is simply 2i times the
sum of the residues at the singular points enclosed by the contour C.
In order for the Residue Theorem to be of much help in evaluating integrals, there needs

to be some better way of computing the residue—finding the Laurent expansion about
each isolated singular point is a chore. We shall now see that in the case of a special but
commonly occurring type of singularity the residue is easy to find. Suppose z is an
0
isolated singularity of f and suppose that the Laurent series of f at z contains only a finite
0
number of terms involving negative powers of z – z . Thus,
0
c c c



f(z)   n   n 1  ...   1  c  c (z z ) ...
0
0
(z z ) n (z z ) n 1 (z z )
1




0
0
0
Multiply this expression by (z – z ) :
n
0
(z) = (z – z ) f(z) = c + c –n+1 (z – z ) + ... + c (z – z ) + ...
n
n–1
–n
0
–1
0
0
What we see is the Taylor series at z for the function (z) = (z – z ) f(z). The coefficient of
n
0
0
(z – z ) is what we seek, and we know that this is
n–1
0

 (n 1) (z )
0
(n 1)!

10.4 Keywords
Singular point: A singular point z of f is said to be isolated if there is a neighborhood of z which
0
0
contains no singular points of f save z .
0
Residue Theorem. It says that the integral of f is simply 2i times the sum of the residues at the
singular points enclosed by the contour C.
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