Page 124 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 12: Fundamental Theorem of Algebra

12.2 Calculus of Residues Notes

The main result to be discussed here is Cauchy’s residue theorem which does for meromorphic
functions what Cauchy’s theorem does for holomorphic functions. This theorem is extremely
important theoretically and for practical applications.

The Residue at a Singularity

We know that in the neighbourhood of an isolated singularity z = a, a one valued analytic
function f(z) may be expanded in a Laurent’s series as

 
n
f(z) =  a (z a)   b (z a)  n


n
n
n 0 n 1


The co-efficient b is called the residue of f(z) at z = a and is given by the formula
1
1
f(z)dz
1
Res (z = a) = b = 2 i   f(z) dz | b n 2 i (z a)  n 1


1




Where g is any circle with centre z = a, which excludes all other singularities of f(z). In case,
z = a is a simple pole, then we have
 b
n
Res (z = a) = b = lim (z-a) f(z) |  a (z – a) + 1
1

z a 0 n z a
A more general definition of the residue of a function f(z) at a point z = a is as follows.
If the point z = a is the only singularity of an analytic function f(z) inside a closed contour C, then
the value f(z) dz is called the residue of f(z) at a.
Residue at Infinity
If f(z) is analytic or has an isolated singularity at infinity and if C is a circle enclosing all its
singularities in the finite parts of the z-plane, the residue of f(z) at infinity is defined by
1 1
Res (z = ) =  f(z) dz, | or Res (z = ) –  f(z) dz,
2 i C 2 i C


Integration taken in positive sense
the integration being taken round C in the negative sense w.r.t. the origin, provided that this
integral has a definite value. By means of the substitution z = w , the integral defining the
-1
residue at infinity takes the form
1  1 dw

2 i  [ f(w )] w 2 ,

taken in positive sense round a sufficiently small circle with centre at the origin.
Thus, we also say if
lim [–f(w ) w ] or lim [–z f(z)]
–1
–1
w 0 w 0
has a definite value, that value is the residue of f(z) at infinity.

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