Page 114 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 10: Residues and Singularities

6. Suppose g is analytic and has a zero of order n at z . Show that the function f given by Notes
0
g'(z)
f(z) 
g(z)
has a simple pole at z , and Res f  n.
0
z 0 z
7. Find :
cosz dz,
2 
C z  4
where C is the positively oriented circle |z| = 6.
8. Find :

 tanzdz,
C
where C is the positively oriented circle |z| = 2.

9. Find :
1
2  dz,

z  z 1
C
where C is the positively oriented circle |z| = 10.
Answers: Self Assessment

1. Singular point 2. Residue Theorem

3. Residue Theorem 4. Laurent series
10.7 Further Readings

Books Ahelfors, D.V. : Complex Analysis
Conway, J.B. : Function of one complex variable
Pati, T. : Functions of complex variable
Shanti Narain : Theory of function of a complex Variable

Tichmarsh, E.C. : The theory of functions
H.S. Kasana : Complex Variables theory and applications
P.K. Banerji : Complex Analysis
Serge Lang : Complex Analysis

H. Lass : Vector & Tensor Analysis
Shanti Narayan : Tensor Analysis
C.E. Weatherburn : Differential Geometry
T.J. Wilemore : Introduction to Differential Geometry

Bansi Lal : Differential Geometry.

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