Complex Analysis - II

Code: MA639 | L-T-P-C: 4-0-0-8

 

Prerequisites: MA547 (Complex Analysis) or equivalent

Preamble: This course is primarily intended for Ph.D. students and can benefit MSc (Mathematics, 3rd Semester onwards, and BTech seventh semester onward), which teaches most fundamental topics at an advanced level. This course is helpful in research for various subjects, like Harmonic Analysis, Complex Analysis, PDE, etc.

Syllabus: Logarithmic function, Branches, Meromorphic Functions, Analytic continuation, Range of Analytic Function, Little Picard's Theorem, Riemann mapping theorem, Boundary Values of Riemann Maps, Phragmén–Lindelöf principle, Boundary behavior of Poisson integral, Harnack's Theorem, Weak compactness principle, Herglotz and Riesz theorem, Non-tangential limit, and Fatou's Theorem, Weierstrass factorization Theorem, Jensen's formula, Blaschke products, Hardy Spaces on the Disk; Identification theorem, Nevanlinna Class, Factorization of Functions in the Nevanlinna Class, Invariant Subspaces, Szego's Theorem

Texts:

1. W. Rudin, Real and Complex Analysis, 3 rd Edn., McGraw Hill, 1987.
2. J. B. Conway, Function of Complex Variables II, GTM, Springer, 1995.

References:

1. J. B. Garnett, Bounded Analytic Functions, GTM, Springer, 2007.
2. B. Ja. Levin, Distribution of Zeros of Entire Functions, Revised Edn., Vol 5, AMS, 1980
3. L. V. Ahlfors, Complex Analysis, 3rd Edn., McGraw Hill, 1979.