**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.

- 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.

# | 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 |