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Indian Institute of Science Education and Research Berhampur

Mathematics

MTH 504: Complex Analysis II (4)

Pre-requisites:
Required: MTH 303 Real Analysis I, MTH 407 Complex Analysis
Desirable: MTH 304 Metric Spaces and Topology, MTH 503 Functional Analysis

Review of elementary concepts: Complex differentiation, Cauchy-Riemann equations, holomorphicity, complex integration, Cauchy’s theorem and Cauchy’s integral formula, Taylor and Laurent series, residue theorem, definition of a meromorphic function.

Harmonic functions: definition and properties, Poisson integral formula, mean-value property, Schwarz reflection principle, Dirichlet problem

Maximum modulus principle: Maximum modulus theorem, Schwarz lemma, Phragmen-Lindelof theorem

Approximations by rational functions: Runge’s theorem, Mittag-Leffler theorem

Conformal mappings: definition and examples, space of holomorphic functions, Montel’s theorem, statement of Riemann mapping theorem

Entire functions, Infinite products, Weierstrass factorization theorem, little and big Picard Theorems, Gamma function

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