Class: TuTh, 10:00-11:30AM, LH-4.
Instructor: Ved Datar
Email: vv lastname at math.iisc.ac.in, no spaces
Office: X05
Office hours: TuTh 11:30-12:30PM, or by appointment (only if you have another class during office hours).
Reference books: Lars Ahlfors, Complex Analysis , McGraw-Hill, 1979.
Supplementary reading: John Conway, Functions of One Complex Variable, Springer-veriag, 1978.
Pre-requisites: MA 221
Topics to be covered: Complex numbers, holomorphic and analytic functions, Cauchy-Riemann equations, Cauchy’s integral formula, Liouville’s theorem and proof of fundamental theorem of algebra, the maximum-modulus principle. Isolated singularities, residue theorem, Argument Principle. Mobius transformations, conformal mappings, Schwarz lemma, automorphisms of the disc and complex plane. Normal families and Montel’s theorem. The Riemann mapping theorem. If time permits - analytic continuation and/or Picard’s theorem.
Homeworks - 20%, Midterms - 30%, Final - 50%
There will be 6 homeworks. The best five will be counted towards the grade. There is no late submission of homeworks.
There will be one midterm of 60 points each and a final exam of 100 points. To pass the class, you have to take the final exam.
| Number | Date | Topic | Homework | Notes |
| 1 | Tu 01/07 | Introduction to complex analysis, complex number system | A1 (due 21/01/20) | Lecture-1 |
| 2 | Th 01/09 | topology of the complex plane | Lecture-2 | |
| 3 | Tu 01/14 | Holomorphicity, power series | Lecture-3 | |
| 4 | Th 01/16 | exponential, trigonometric functions and logarithm | Lecture-4 | |
| 5 | Tu 21/01 | Cauchy-Riemann equations | A2 (due 04/02/20) | Lecture-5 |
| 6 | Th 23/01 | Complex integration | Lecture-6 | |
| 7 | Tu 28/01 | Cauchy's and Grousat's theorems | Lecture-7 | |
| 8 | Th 30/01 | Cauchy Integral formula, Analyticity | Lecture-8 | |
| 9 | Tu 04/02 | fundamental theorem of algebra, Zeroes of holomorphic functions, sequences | Lecture-9 | |
| 10 | Th 06/02 | Index, Generalized Cauchy theorem | A3 (due 25/02/20) | Lecture-10 |
| 11 | Tu 11/02 | Cauchy's theorem on simply connected and multiply connected domains | Midterm Practice Problems | Lecture-11 |
| 12 | Tu 13/02 | logarithms, roots and branch cuts | Lecture-12 | |
| Tu 18/02 | midterm | |||
| 13 | Tu 25/02 | Special lecture - Riemann surfaces | ||
| 14 | Th 27/02 | isolated singularities - removables singularities, poles and essential singularities | A4 (due 10/03/20) | Lecture-14 |
| 15 | Tu 03/03 | Laurent series expansions | Lecture-15 | |
| 16 | Th 05/03 | Meromorphic functions | Lecture-16 | |
| 17 | Tu 10/03 | Theroems of Mittag-Leffler and Weierstrass | Lecture-17 | |
| 18 | Th 12/03 | Residue theorem and argument principle | A5 (due 31/03/20) | Lecture-18 |
| 19 | Tu 17/03 | Applications of argument principle - Rouche's theorem, open mapping theorem, max modulus principle | Lecture-19 | |
| 20 | Th 19/03 | real variable integrals - I | Lecture-20 | |
| 21 | Tu 24/03 | real variable integrals - II | Lecture-21 | |
| 22 | Th 26/03 | conformal maps | Lecture-22 | |
| 23 | Tu 31/03 | Automorphisms of the plane, The Riemann mapping theorem | Lecture-23 | |
| 24 | Th 02/04 | Proof of the Riemann mapping theorem | Lecture-24 | |
| 25 | Tu 07/04 | The Gamma and Rieman Zeta functions | Lecture-25 |