Details of MA4101 (Autumn 2012)
| Level: 4 | Type: Theory | Credits: 3.0 |
| Course Code | Course Name | Instructor(s) |
|---|---|---|
| MA4101 | Complex Analysis | Sriram Balasubramanian |
| Syllabus |
|---|
| MA4101 : Advanced 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: Analytic functions, 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. Space of Analytic Functions: Space of analytic functions, normal family, normality and compactness, Riemann mapping theorem. Analytic Continuation: Schwarz's reflection principle, analytic continuation along a path, Monodromy theorem. Runges Theorem: Runge's approximation theorem, Mittag-Lefflers theorem. Special Functions: Beta and Gamma functions, Riemann zeta functions. Harmonic Functions: Basic properties of harmonic functions, resolution of Dirichlet problem on unit disc. Range of an Analytic Function: Little Picard theorem, great Picard theorem. |
| References |
|---|
| 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. | Programme | Semester No | Course Choice |
|---|---|---|---|
| 1 | IP | 1 | Not Allowed |
| 2 | IP | 3 | Not Allowed |
| 3 | MS | 7 | Not Allowed |
| 4 | RS | 1 | Not Allowed |