Details of MA4101 (Autumn 2019)

Level: 4 Type: Theory Credits: 4.0

Course CodeCourse NameInstructor(s)
MA4101 Complex Analysis Sushil Gorai

MA4101 :Complex Analysis

The Complex Number System: The field of complex numbers, polar representations, power, roots, complex exponential, complex logarithm, extended complex plane, Riemann sphere and stereographic projection.

Analytic Functions: Definitions, Cauchy-Riemann equations, harmonic functions, conformal maps and geometry of Mbius transformations.

Complex Integration: Riemann-Stieltjes integration, power series representation of analytic functions, zeros of an analytic function, winding number, Cauchys integral formula, Cauchy estimates and Liouville theorem, Cauchys theorem, Morera's theorem, open mapping theorem, maximum modulus theorem, Schwarzs lemma.

Singularities: Classification of singularities, Laurent series, Casorati-Weierstrass theorem, residues, evaluation of definite integrals using residue theorem, meromorphic functions, the argument principle, Rouchs theorem.

Conformal Mappings: Definitions, normal family, normality and compactness, Riemann mapping theorem.

Analytic Continuation: Schwarz's reflection principle, analytic continuation along a path, Monodromy theorem, Little Picard theorem.

Analysis III (MA3101), Topology (MA3201).

Suggested Texts/Reference Books:

1. Ahlfors, L. V., Complex Analysis (3rd Edition), McGraw-Hill, 1979.

2. Conway, J. B., Functions of One Complex Variable (2nd Edition), Springer-Verlag, 1978.

3. Greene, R. E. and Krantz, S. G., Function Theory of One Complex Variable (3rd Edition), Graduate Studies in Mathematics, Volume 40, American Mathematical Society, 2006.

4. Lang, S., Complex Analysis (4th Edition), Graduate Text in Mathematics, Vol 103, Springer-Verlag, 1999.

5. Marsden, J. E., Basic Complex Analysis, W. H. Freeman and Co., 1973.

6. Narasimhan, R., Complex Analysis in One Variable, Birkhauser-Verlag, 1985.

7. Rudin, W., Real and Complex Analysis (3rd Edition), McGraw-Hill, 1987.

Course Credit Options

Sl. No.ProgrammeSemester NoCourse Choice
1 IP 1 Not Allowed
2 IP 3 Core
3 IP 5 Not Allowed
4 MR 1 Not Allowed
5 MR 3 Not Allowed
6 MS 7 Core
7 RS 1 Elective
8 RS 2 Elective