**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

*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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