Lecture Format and Access to Resources. Lectures will be held on Microsoft Teams. All the resources related to the course (recordings of the lectures, notes, assignments, etc.) will be posted on the MA 224 team page. If you wish to attend this course, please add yourself to the MA 224 team using the team link/code available on the IISc intranet bulletin board.
Instructor. Purvi Gupta (purvigupta(at)iisc(dot)ac(dot)in)
Tutor/TA. Pritam Ganguly (pritamg(at)iisc(dot)ac(dot)in)

Lectures. MWF 2:00 - 3:00 pm.
Tutorials. M 6:00 - 7:00 pm (first session: Mar. 08).
Office hours. TBA.

Course Description. This course will be an introduction to complex analysis in one variable. We will cover the standard topics (listed below) for a course at this level. If time permits, we will cover one or two topics in addition to the ones listed below.

Basic properties of complex numbers. Complex differentiation, holomorphicity and the Cauchy—Riemann equations. Complex integration, Cauchy’s theorem and integral formula, power series representability, and Liouville’s theorem. Simply connected domains. Isolated singularities, residues, and the argument principle. The open mapping theorem, the maximum modulus principle and Rouché’s theorem. Conformal mappings, Schwarz’s lemma, automorphisms of the disc and the complex plane. Normal families and Montel’s theorem. The Riemann mapping theorem.

Suggested References.
  • L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979.
  • J. B. Conway, Functions of One Complex Variable, Springer-Verlag, 1978.
  • For more practice problems: Complex Analysis by T. W. Gamelin and/or Complex Analysis by E. M. Stein and R. Shakarchi.
Evaluation Scheme.
Course Calendar. Typically, I will update the calendar twice a week --- on Saturdays (to update the upcoming week's plan), and on Wednesdays (to post assignments).

# Date (Day) Topics Assignment Tracker
Week 1
1. 22/02 (M) Basic properties of the complex plane
2. 24/02 (W) Topological properties of the plane; C-linearity Assignment 1
3. 26/02 (F) Holomorphicity and analyticity
Week 2
4. 01/03 (M) Examples of holomorphic functions
5. 03/03 (W) Examples (contd.) and conformality Assignment 2
6. 05/03 (F) Complex Integration
Week 3
7. 08/03 (M) Primitives and Cauchy's theorem
08/03 (M) Tutorial 1 Quiz 01
8. 10/03 (W) The missing piece: Goursat's argument
9. 12/03 (F) Cauchy's integral fomula Assignment 3
Week 4
10. 15/03 (M) The Cauchy--Pompeiu integral formula
15/03 (M) Tutorial 2 Quiz 02
11. 17/03 (W) Applications of the Cauchy integral formula
12. 19/03 (F) Applications (contd.)
Week 5
13. 22/03 (M) The winding number
22/03 (M) Tutorial 3 No quiz.
14. 24/03 (W) The global form of Cauchy's theorem Assignment 4
15. 26/03 (F) The generalized Cauchy integral formula
Week 6
16. 29/03 (M) Simply connected planar sets
29/03 (M) Tutorial 4 Quiz 03
17. 31/03 (W) Simple-connectivity (contd.)
02/04 (F) No lecture (Good Friday) Assignment 5
Week 7
18. 07/04 (M) Isolated singularities
05/04 (M) Tutorial 5 Quiz 04
19. 07/04 (W) Laurent series expansions
20. 09/04 (F) The residue theorem
Midterm week
12/04 (M) Tutorial 6 No quiz.
16/04 (F) Midterm Exam (2-3:30 pm)
Week 8
21. 19/04 (M) The residue theorem (contd.)
19/04 (M) Tutorial 7 No quiz
22. 21/04 (W) Contour integration
23. 23/04 (F) More contour integration Assignment 6
Week 9
24. 26/04 (M) The argument principle
26/04 (M) Tutorial 8 Quiz 05
25. 28/04 (W) Applications of the argument principle
26. 30/04 (F) Applications of the argument principle (contd.)
Week 10
03/05 (M) Class cancelled.
03/05 (M) Tutorial 9 No quiz
27. 05/05 (W) Conformal mappings: examples
28. 07/05 (F) Automorphism groups of the plane, extended plane and the unit disk Assignment 7
Week 11
10/05 (M)
10/05 (M) Tutorial 10 Quiz 6