ADVANCED COMPLEX ANALYSIS (Optional Paper)
Conformal mapping, bilinear transformation mappings, Special mappings:
W(z)= 1/z, z+1/z, z2, ez , Sin(z),Cos(z)
Power Series: Absolute convergence, Cauchy’s Hadamard theorem, Circle and radius of convergence, Analyticity of the sum function or a power series, Complex inversion formula for inverse Laplace transform and related problems.
Schwarz’s lemma and its consequences, Doubly periodic entire functions, Fundamental theorem of algebra, Zeros of certain polynomials.
Meromorphic functions, Essential singularities and Picard’s theorem, Analytic continuation, Monodrmy theorem, Poisson integral formula, Analytic continuation via reflexion.
Infinite sums and infinite product of complex numbers, Infinite product of analytic functions, Factorization of entire function.
1. S. Ponnusamy, Foundation of Complex Analysis, Narosa Publishing House, 2011.
2. L. R. Ahlofrs, Complex Analysis, Mc-Graw Hill, 1979.
3. A.S.B. Holland, Introduction to theTtheory of Entire Functions, Academic Press, 1973.
4. H.S. Kasana, Complex Variables: Theory and Applications, Prentice-Hall, New Delhi, 2005.
1. Mark J. Ablowitz, A.S. Fokas, Complex Variables: Introduction and Applications, Cambridge University Press South Asian Edition, 1998.
2. J.W. Brown, R.V. Churchil, Complex Variables and Applications, McGraw Hill, New York, 1990.
3. R. Murray Spiegel, Theory and Problems of Complex variables, Schaum Outline Series, 1974.
4. K.K. Dubey, Fundamentals of Complex Analysis Theory and Applications, International Publishing House, 2009.
5. Joseph Bak, Donald J. Newman, Complex Analysis, Springer, 2010.
