Page 141 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 3. 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 residue at infinity takes the form ................
-1
taken in positive sense round a sufficiently small circle with centre at the origin.
4. If the function f(z) has a simple pole at z = a, then, Res (z = a) = ................
5. If f(z) is analytic except at a finite number of singularities and if f(z)0 uniformly as z,
then ................, where T denotes the semi-circle |z| = R, I . z  0, R being taken so large that
m
all the singularities of f(z) lie within T.
6. a function f(z) is said to be single-valued if it satisfies f(z) = f(z(r, )) = f(z(r,  + 2))
otherwise it is classified as ................
7. A ................ is a portion of a line or curve that is introduced in order to define a branch F(z)
of a multiple-valued function f(z).
8. A ................ f(z) defined on S   is said to have a branch point at z when z describes an
0
arbitrary small circle about z , then for every branch F(z) of f(z), F(z) does not return to its
0
original value.
12.8 Review Questions

1. Discuss the concept of fundamental theorem on algebra.

2. Describe the calculus of residues.
3. Discuss the multivalued functions and its branches.

Answers: Self Assessment

1. n zeros. 2. inverse function of f.

1 1 dw
3.  [ f(w )] , 4. lim (z–a) f(z)


2 i w 2 z a

5. lim e imz f(z) dz = 0 , m > 0 6. multivalued function.

R
T
7. branch cut 8. multivalued function
12.9 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

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