MA3B8 Complex analysis
Term I 2025-2026

Module Description

In this module, MA3B8 (complex analysis), we assume as background the material from the second year core. We will particularly need some point-set topology in the plane, one-variable real analysis, and the machinary of power series.

Complex analysis is calculus in one variable, but where that variable ranges over the complex numbers instead of the reals. This change transforms the subject; the objects in complex analysis have a new and beautiful rigidity very unlike their real siblings. One example of this is the capstone result of the course - Riemann's mapping theorem. Along the way, we will cover holomorphic functions, Cauchy's theory of contour integrals, Weierstrass's theory of analytic functions, the residue theorem and its applications to improper definite real integrals, and Riemann's theory of conformal mappings.

Schedule

The schedule will contain the list of topics (organised by week and lecture). We will update this as we work through the material. Links to (handwritten) lecture notes and (typed) example sheets will be posted week-by-week. Recorded lectures are available via lecture capture.

Instructor, teaching fellow, and TA

Class meetings

Reference materials

I am writing lecture notes for the module. I have written solutions for all (but one) of the exercises in the lecture notes; the solutions can be found in Appendix A. If you have questions about anything in the notes, or find any errors, or find possible improvements, then please inform me via the Q+A forum, via the anonymous form, or via email (if there is no other way). Previous versions of the module were lectured by Peter Topping and Mark Pollicott. We will not follow their syllabi in every detail; however, our choice of topics will be very similar. Their lecture notes, from previous versions of the module can be found here: Topping and Pollicott.

Other useful references for the material include the following:

• Complex analysis: an introduction to the theory of analytic functions of one complex variable by Lars Ahlfors;
• Theory of complex functions by Reinhold Remmert; and
• Complex function theory by Donald Sarason;

Links to the lecture capture, the announcement forum, and the Q+A forum are on the module's Moodle page.

Example sheets

See the schedule for the example sheets. These will not be assessed. Unlike the exercises in the lecture notes, I have not prepared worked solutions for the exercises in the example sheets. Please let me (Saul) know, using the Q+A forum, if any of the problems are unclear or have typos.

Exam

The exam will be 100% of your mark. The exam will be closed book. Here are the exam papers for this module from the last five years.

Mistakes

Please inform me of any errors on this website, on the Moodle page, or made in class. You can use the Q+A forum for this. You can also tell me in, or immediately after, lecture, or in office hours, or via email.