MATH GU4065

Review of basic properties of complex number. Definition of holomorphic and entire function. Cauchy-Riemann Equations. Goursat's Theorem and Cauchy's Integral Theorem. Construct contours to solve complex integrals and improper integrals. Definition of Gamma Function and Riemann Zeta Function. Euler's Formula of gamma function. Relation between zeta function and prime numbers. Arzela-Ascoli theorem and Riemann Mapping Theorem.

The review of him below in Honors Complex Analysis class made me wonder if the person actually took the same class with me. Maybe he's trying to trick people. His writing is ineligible and the lecture is impossible to understand even on zoom. Believe me, I rewatched his lecture and still, there are a fair amount of parts that I couldn't understand what he is saying. He is obsessed with big O notation and gives very hands-wavy proof, which otherwise would be treated more rigorously. The practice problem sets in Shakarchi-Stein are terrible in my view and didn't definitely help me in properly training for basic concepts. I honestly skipped 1/3 of the lectures because I found no point in it. Also, it was very clear that he doesn't give a damn about this class

Courseload is higher than an average math class, though that may be expected given this is a rigorous introduction to complex analysis + low-dimensional riemann surfaces: Grade breakdown: HW: 30% of grade; 1 per week Midterm: 30% of grade; 1 per term Exam: 40% of grade (cumulative, emphasis on what’s instructed in latter half of term) Material covered: 1. Holomorphic functions 2. Cauchy-goursat theory (local/global C-G theory) 3. Laurent series (singularities, residue theorem, argument principle) 4. Examples of constructing meromorphic functions (gamma function, zeta function, entire functions) 5. Introduction to Riemann surfaces (complex dynamics, conformal mappings, harmonic functions, Riemann mapping) HW took anywhere from 12-30 hours per week to complete, but given that this is an honors class that is rather expected. The courseload is manageable but it requires dedication, and a strong understanding of foundational real analysis. It is worth mentioning that Tang-Kai is very happy to discuss the necessary preparation required to succeed in Honors Complex Variables, as well as putting in a lot of time out of class to help students fill in the necessary gaps to stay on top of the material.

I learned so many theorems from this class that I was shocked. I think these theorems represent the purest beauty about mathematics. This is something I didn't expect at all from the name of the class - complex variables.

The material of complex analysis is a fun counterpoint to what you learn in the real analysis sequence. Professor Dubédat is clear in lectures, and the textbook by Stein and Shakarchi is also quite nice for self-study (esp. in comparison to Rudin). The course moved at a leisurely pace (we only covered Ch. 1-3,7,9 + some of the appendix in the book) but this allowed for plenty of time to understand the material, which is a good thing since many of the basic results in complex analysis are quite beautiful. Overall in an underwhelming semester of Covid-uni this was a highlight. If you're a math major, definitely take this course rather than the 3000-level one, which is largely lacking in rigor.

Like many other students, I have had several professors throughout my journey in higher mathematics, and I have never met somebody quite as polite, committed, and supportive as Professor Tang-Kai. Honors Complex Variables is objectively a difficult class, and despite the hurdles that come with teaching such a difficult course, Tang-Kai did an incredible job despite it being his first year teaching at a university. He taught the material so naturally (as in clean examples, very thorough responses to questions that correctly gauge the student’s current level of understanding) with clean structure and fluidly that there almost wasn’t even a need for him to look at his lecture notes for reference. Generally, the classroom environment was very nice, this class in particular was held in the morning, and when I didn’t attend class Tang-Kai would reach out to me personally via email as well as the next in-person class session just to check in on me. Of course, this would be moot had it been via special selection, but the truth is that Tang-Kai checked in with ALL of his students rather consistently. In addition to this, he wrote very detailed lecture notes for Honors Complex Variables, and his ability to write at such a high-level’s worth of mathematical exposition serves as a critical demonstration of his ability for teaching excellence. The classroom started with around 6-7 people,…

Content-wise, we covered the standard material in a typical complex analysis class—holomorphic functions up to the residue theorem. After that, we talked a little bit about simply-connectedness and how that was related to the logarithm, the gamma/zeta function (with a little hint at how the zeta function was related to number theory), conformal mappings, and the Riemann mapping theorem.

A comprehensive introduction to complex analysis. Cauchy's integral formula, residue theorem, conformal mappings, etc.

Don't look at the name of this class, complex variables, it's simply beauty. Countless mathematicians have been shocked by its contents.

The material covered in this class is really interesting but the style of lectures made it really difficult to learn. Take this class with a different instructor. 1 Columbia University: Arts & Sciences Fall 2022 Course: MATHGU4065_001_2022_3-HONORSCOMPLEXVARIABLES : MATHGU4065_001_2022_3 - HONORS COMPLEX VARIABLES Instructor: Francesco Lin