# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

## MAT 205B: Complex Analysis

**Approved:**2021-11-02, Hunter/Romik

**Units/Lecture:**

4 units

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

Complex Analysis, by E. M. Stein and R. Shakarchi (Princeton University Press, 2003)

**Prerequisites:**

MAT 205A

**Course Description:**

A second graduate course in complex analysis covering some of the more advanced topics in the theory. Specific topics will vary depending on the instructor’s preference.

**Suggested Schedule:**

Any combination of the following topics is suitable to cover; suggested references appear in parentheses:

- Conformal maps and the Riemann mapping theorem (Chapter 8 in [5])
- Elliptic functions (Chapter 9 in [5])
- Modular forms and theta functions; applications to number theory (Chapter 10 in [5])
- The Fourier transform in complex analysis (Chapter 4 in [5])
- Entire functions and the Weierstrass-Hadamard theory of infinite products (Chapter 5 in [5])
- Introduction to Riemann surfaces ([2], [3])
- Other topics at the instructor’s discretion

The advanced topics covered in MAT205A and MAT205B do not need to be learned in a specific order. In a given year, the instructors teaching those classes may decide to cover some of the advanced topics suggested for MAT205B in MAT205A, and vice versa.

REFERENCES:

- M. J. Ablowitz, A. S. Fokas. Complex Variables: Introduction and Applications, 2nd Ed. Cambridge University Press, 2003.
- R. Narasimhan, Y. Nievergelt. Complex Analysis in One Variable, 2nd Ed. Birkhauser, 2001.
- O. Forster. Lectures on Riemann Surfaces. Springer, 1981.
- D. Romik. Complex Analysis Lecture Notes (version of June 15, 2021). Free online resource.
- E. M. Stein, R. Shakarchi. Complex Analysis (Princeton University Press, 2003).