Page 110 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 10: Residues and Singularities

This is now easy. The contour is positive oriented and encloses two singularities of f; viz, Notes
i and –i. Hence,

e z

2  dz = 2 i Res f Res f 


2
z (z  1)  
C  z 0 z 
i

i 
 e i e 
= 2 i   

 2i 2i 
= –2i sin 1.
There is some jargon that goes with all this. An isolated singular point z of f such that the
0
Laurent series at z includes only a finite number of terms involving negative powers of z – z is
0
0
called a pole. Thus, if z is a pole, there is an integer n so that (z) = (z – z ) f(z) is analytic at z ,
n
0
0
0
and f(z )  0. The number n is called the order of the pole. Thus, in the preceding example, 0 is a
0
pole of order 2, while i and –i are poles of order 1. (A pole of order 1 is frequently called a simple
pole.) We must hedge just a bit here. If z is an isolated singularity of f and there are no Laurent
0
series terms involving negative powers of z – z , then we say z is a removable singularity.
0
0
Example:
Let
sinz
f(z)  ;
z
then the singularity z = 0 is a removable singularity:
1 1 z 3 z 5
f(z) = sinz  (z    ...)
z z 3! 5!
z 2 z 4
= 1    ...
3! 5!
and we see that in some sense f is “really” analytic at z = 0 if we would just define it to be the right
thing there.
A singularity that is neither a pole or removable is called an essential singularity.
Let’s look at one more labor-saving trick—or technique, if you prefer. Suppose f is a function:

p(z)
f(z)  ,
q(z)

where p and q are analytic at z0, and we have q(z ) = 0, while q’(z )  0, and p(z )  0.
0
0
0
Then,
p(z) p(z ) p'(z )(z z ) ...



f(z)   0 0 n 0
q(z) q (z ) 2
0
q'(z )(z z )  2 (z z ) ...


0
0
0

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