Page 120 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 11: Rouche’s Theorem

5. Suppose f is entire and f(z) is real if and only if z is real. Explain how you know that f has Notes
at most one zero.
6. Show that the polynomial z + 4z – 1 has exactly two zeros inside the circle |z| = 1.
2
6
7. How many solutions of 2z – 2z + 2z – 2z + 9 = 0 lie inside the circle |z| = 1?
2
4
3
8. Use Rouche’s Theorem to prove that every polynomial of degree n has exactly n zeros
(counting multiplicity, of course).
9. Let C be the closed unit disk |z|  1. Suppose the function f analytic on C maps C into the
open unit disk |z| < 1—that is, |f(z)| < 1 for all z  C. Prove there is exactly one w  C such
that f(w) = w. (The point w is called a fixed point of f .)

Answers: Self Assessment

1. singularities 2. positive direction
3. Y(0) = Y(1)

11.7 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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