# 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 205A: 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:**

MAT185A (Complex Analysis) or equivalent

**Course Description:**

A first graduate course in complex analysis. The course's
goal is to develop the theory in a rigorous and systematic fashion and introduce the students to the main concepts and results of the topic, notably contour integration, Cauchy's theorem, and the interplay between complex analytic ideas and the topology of planar curves. Some advanced topics will also be covered.

**Suggested Schedule:**

- The basic theory (Chapters 1–3 in [5])
- Analyticity, Cauchy-Riemann equations, conformality of analytic maps, connection to harmonic functions
- Contour integrals
- Homotopy of curves and simple-connectedness
- Goursat’s theorem, Morera’s theorem, Cauchy’s theorem, Cauchy integral formula
- Liouville’s theorem and the fundamental theorem of algebra
- Zeros and poles. Classification of singularities, the Riemann removable singularity theorem and the Casorati-Weierstrass theorem
- Principle of analytic continuation
- Taylor and Laurent series
- The residue theorem
- The logarithm and power functions
- Winding numbers, the argument principle, Rouché’s theorem, and applications

- Advanced topics
- The Euler gamma function (Chapter 6 in [5]
- The Riemann zeta function (Chapter 6 in [5])
- The prime number theorem (Chapter 7 in [5]; see also [4, Sec. 16])
- Introduction to asymptotic analysis (Appendix A in [5]; see also [4, Sec. 17] and [1, Chapter 6]

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).