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
- The basic theory (Chapters 1–3 in )
- 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 
- The Riemann zeta function (Chapter 6 in )
- The prime number theorem (Chapter 7 in ; see also [4, Sec. 16])
- Introduction to asymptotic analysis (Appendix A in ; 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.
- 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).