Page 125 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Remarks. (i) The function may be regular at infinity, yet has a residue there.

b
For example, consider the function f(z) = for this function
z a

1
Res (z = ) =   f(z) dz
2 i C

1 b

=  2 i z a dz



C
b 2 re id
i

= – 2 i 0  re i , C being the circle |z – a| = r


b 2
=  2 0  d   b

 Res (z = ) = –b
Also, z = a is a simple pole of f(z) and its residue there is 1  f(z) dz = b
2 i C

| or lim (z–a) f(z) = b
z a
Thus, Res (z = a) = b – Res (z = )
(ii) If the function is analytic at a point z = a, then its residue at z = a is zero but not so at
infinity.
(iii) In the definition of residue at infinity, C may be any closed contour enclosing all the
singularities in the finite parts of the z-plane.

Calculation of Residues
Now, we discuss the method of calculation of residue in some special cases.

(i) If the function f(z) has a simple pole at z = a, then, Res (z = a) = lim (z–a) f(z).
z a

(ii) If f(z) has a simple pole at z = a and f(z) is of the form f(z) =  (z) i.e. a rational function,
 (z)
then

Res (z = a) = lim (z-a) f(z) = lim (z-a)  (z)
z a z a  (z)

(z)

= lim a  (z)   (a)

z
z a

=  (a) ,
 '(a)

where (a) = 0, (a)  0, since (z) has a simple zero at z = a

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