MA5360 Complex Analysis
Unit I: Topology of the Complex plane, limits, continuity, Complex functions – Introduction to the complex exponential, the complex logarithm and trigonometric functions, Linear fractional transformations. (9 Lectures).
Unit II: Differentiability, Cauchy-Riemann equations, Sufficient conditions for differentiability, holomorphic functions, harmonic functions, harmonic conjugates, branches of the logarithm. (9 Lectures)
Unit III: The complex integral, Cauchy's theorem, The Cauchy integral formula, Morera's theorem, Liouville's theorem, Maximum Modulus Principle, Schwarz lemma, Open mapping theorem, Convergence of series of complex numbers, Power series, Radius of convergence, Differentiation of power series, absolute and uniform convergence of a power series, Taylor’s Series, Laurent’s Series, Singularities, Classification of singularities, Cauchy’s Residue theorem, Behaviour of a function near singularities including the CasoratiWeierstrass theorem, evaluation of Real integrals, Argument principle, Rouche’s theorem. (22 Lectures).
1. S. Ponnusamy and H. Silverman, Complex Variables with Applications, Birkhauser, Boston, 2006.
2. D. Sarason, Complex Function Theory, Hindustan Book Agency, 2008.
1. L. Ahlfors, Complex Analysis, 2nd ed., McGraw-Hill, New York, 1966.
2. T.W. Gamelin, Complex Analysis, Springer-Verlag, 2001.
3. J.B. Conway, Functions of one Complex Variable, 2nd edition, Springer-Verlag, 1978.
4. R.E. Greene and S.G. Krantz, Function theory of One Complex variable, AMS, 2006.
5. J.W. Brown and R.V. Churchill, Complex Variables and Applications, McGraw Hill, 2008.