## 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 205: Complex Analysis

**Approved:**2007-09-01, Michael Kapovich

**Units/Lecture:**

4 units

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

[SS] Elias Stein and Rami Shakarachi Complex Analysis, Princeton Lectures in Analysis. [See below for supplementary texts.]

Search by ISBN on Amazon: 978-0-691-11385-2

Search by ISBN on Amazon: 978-0-691-11385-2

**Prerequisites:**

MAT 185A (Complex Analysis)

**Course Description:**

Analytic continuation, Riemann mapping theorem, elliptic functions, modular forms, Riemann zeta function, Riemann surfaces.

**Suggested Schedule:**

*Refer to short names of books in [brackets] below.*

- Brief review of complex analysis: Cauchy theorem and integral formula, power series, Cauchy-Riemann equations, harmonic functions. Residue calculus. Argument principle. (Chapters 1, 2, 3 of [SS].)
- Homotopy of paths, the fundamental group. The fundamental group of punctured complex plane, the winding number and the integral of 1/z. (Chapter 3, Appendix B of [SS].)
- Analytic continuation and multi-valued analytic functions. The monodromy principle. (Chapter 6 of [MH].) Riemann surfaces: A 1-dimensional complex manifold defined via an atlas of charts. Examples: the Riemann sphere, Riemann surfaces of multi-valued algebraic and analytic functions. Covering spaces and the universal cover. (Chapter 1 of [M]. Chapter 16, Section 7 of [G].)
- Conformal mappings. Examples. Reflection Principle. (Chapter 2, Section 5.4 of [SS].) Schwarz lemma. Normal families. Riemann mapping theorem. (Chapter 8, Sections 1, 2, 3 of [SS].) Classification of Riemann surfaces (without proof). (Chapter 16 of [G].)
- Entire functions. Weierstrass and Hadamard product theorems. (Chapter 7 of [SS].)
- Special functions. Gamma and zeta functions. Connections with the prime number theorem (sketch of the proof). (Chapters 6, 7 of [SS].)
- Elliptic functions. Weierstrass P-function. (Chapter 8 of [SS].) Interpretation of elliptic functions as meromorphic functions on the torus. (Chapter 5 of [FB].)

**Additional Notes:**

SUPPLEMENTARY TEXTBOOKS

- [M] Rick Miranda, Algebraic Curves and Riemann Surfaces (AMS, Graduate Studies in Mathematics, Vol 5), ISBN-10: 0821802682. (For the detailed treatment of Riemann surfaces.)
- [G] T. Gamelin Complex Analysis, Springer Verlag, 2001, 478 pages, ISBN 0387950699. (For the treatment of classification of Riemann surfaces.)
- [FB] Eberhard Freitag, Rolf Busam, Complex Analysis, Springer Verlag, 2005, 547 pages. ISBN 3540257241. (For the detailed treatment of elliptic functions.)
- [MH] Jerrold E. Marsden, Michael J. Hoffman, Basic Complex Analysis, 1998. ISBN-10: 071672877X