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Indian Institute of Science Education and Research Bhopal
MTH 604: 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
Suggested Books:
Texts:
- Stein E.M. and Shakarchi R., Complex Analysis (Princeton Lectures in Analysis Series, Vol. II), Princeton University Press, 2003
- Conway J.B., Functions of One Complex Variable, Springer-Verlag NY, 1978
- Rudin W., Real and Complex Analysis, McGraw-Hill, 2006
- Epstein B. and Hahn L-S., Classical Complex Analysis, Jones and Bartlett, 2011
- Ahlfors L., Complex Analysis, Lars Ahlfors, McGraw-Hill, 1979.
References:
- Carathodory C., Theory of functions of a complex variable, Vol. I and II, Chelsea Pub Co, NY 1954
- Remmert R., Classical topics in complex function theory, Springer 1997
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