Page 127 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes
z 4

or (z) 
4
4

= lim z  a z ia
4

z ia z ia  2ia z /(z ia)
as f(z) 
(z ia)

a i ia 3
4
= 
2a 2
(b) Find the residues of e z at its poles.
iz
-4
e iz
Solution. Let f(z) =
z 4
f(z) has pole of order 4 at z = 0, so
iz 
Res (z = 0) = 1   d 3 3 (e )    i | (z) = e iz
3 dz
|   z 0 6

Alternatively, by the Laurent’s expansion
e iz 1 i 1 i ...
z 4  z 4  z 3  | 2z 2  | 3z 

we find that

1
Res (z = 0) = co-efficient of
z
i
= –
6

z 3
(c) Find the residue of at z = .
2
z  1

Solution. Let f(z) = z 3  z 3  z 1  1   1

2 
2
z  1 2  1   z 
z  1  2 
 z 
1 1
= z(1    ....)
z 2 z 4
1 1
= z    ...
z z 3
Therefore,

1
Res (z = ) = –(co-efficient of )   1
z
(d) Find the residues of at its poles.

z 3
Solution. Let f(z) =
(z 1) (z 2)(z 3)
4



120 LOVELY PROFESSIONAL UNIVERSITY