MTH403A - Complex Analysis

Topology on C, Convergence and continuity. Cauchy Riemann equation, Elementary Functions. Power series: Convergence, Exponential, Trigonometric functions. Integration along curves, Cauchy Goursat Theorem, Cauchy's theorem for disc, Evaluation of some integrals, Cauchy integral formula, Liouville theorem and fundamental theorem of Algebra, Identity theorem, Morera's theorem. Zeros and poles, Residue theorem, Evaluation of some integrals. Riemann theorem on removable singularities, Essential singularities, Casorati Weierstrass theorem. Riemann sphere, Argument principle, Rouche's theorem, Open mapping theorem, Maximum modulus principle, Cauchy's theorem for simply connected domain, Analyticity of complex logarithm. Harmonic functions, Poisson integral formula, Characterization of harmonic functions through MVP. Fractional linear transformation, Schwartz lemma, Pick's lemma, Auto morphemes of disc and upper half plane. Montel theorem, Riemann mapping theorem.