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# NUS Math Module Review: MA3111S Complex Analysis I (S)

Assessment Structure:

20% Homework (4 sets total)

20% Midterm (In-Class)

60% Final (Online)

This is one of the five S modules, and also usually the last one which SPM students would take. The module’s topics include metric spaces and topologies, and MA3111 Complex Analysis I contents (complex and contour integration, Cauchy’s theorem and its consequences, power series and Taylor’s theorem, Laurent series), MA4247 Complex Analysis II contents (Identity theorem, Riemann’s removable singularity theorem, Casorati-Weierstrass theorem, maximum modulus theorem, winding numbers, Residue theorem, Rouche’s theorem). Additional topics include sequences and series of holomorphic functions, complex manifolds and distribution theory (only first is tested). The notes of reference prior to additional topics is Further Analysis, W. T. Gowers.

Similar to other S modules, class size is relatively small so there is no webcast for live lessons. Lecture and tutorial attendances are not compulsory. Tutorial sessions usually involves Prof. Dinh presenting the solutions, but he actively engages the students to think about how to approach the problem from various perspectives, and how to solve if the hypothesis was slightly changed. Tutorial questions are generally very difficult, and it appears that many students spend a lot of time on tutorial questions but end up not finishing them. Regardless, they are extremely important as this module is a lot about knowing many different tricks to solve problems. Furthermore, many concepts not covered in the lecture notes but tested in the exams originate from tutorial questions.

Concept wise, this module is of average-moderately difficult difficulty. Complex analysis requires a lot of heuristics and intuitive understanding, for otherwise the proofs may look very cryptic. Do not expect to understand everything in this module. Many theorems in the later part of the module require theorems/lemmas from the earlier part, so I strongly advise students to understand the concepts taught (or at least be really familiar with it). It took me a long time before I could appreciate the Cauchy integral formula, in which then many other theorems at the back become clear. As expected of an S module, one should not have time to slack off in this module.

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