Analysis II: Complex Analysis

Description of Course Goals and Curriculum

Complex analysis begins by defining the derivative for complex numbers: we require that the usual limit is the same when taken in any direction in the complex plane. With this rather innocuous step, we arrive at powerful and sometimes disturbing theorems, such as rigidity, analyticity, and the Cauchy Integral Formula. Moving forward in the class, we discuss applications. Most notably: Fourier Transforms, Contour Integrals for proving identities, Prime Number Theorem, Gamma Function, Zeta Function, Conformal Maps, and Number Theory. A large part of the course, and many of the results in the book and on exams, is doing estimates just strong enough for what you’re trying to prove: only do what you have to! Along the same line, you should think from the outset about what you really are trying to prove and what is the bare minimum necessary to do so. Many times, we could use elementary (think Math Olympiad) methods to solve a problem. But the beauty of complex analysis is doing things that could theoretically be done with elementary methods, but doing so easily by using more abstract ideas like contour integrals and rigidity (the basis for the proofs of many of the applications mentioned above). It truly turns pages into sentences. The key to exploiting this reduction is thinking about things pictorially and also making connections to the proofs given along the way. There are a finite number of “tricks” used in the proofs shown in Stein that you too must exploit in homework and on exams. For example, when there are poles/zeros, divide by another function with poles/zeros of the same multiplicity at the same spots and you’ll get an entire, bounded function. Liouville’s Theorem says this is a constant and you’ve found an easy relationship between functions.

Learning From Classroom Instruction

The lectures mostly follow the book, almost identically. The style instruction is very straightforward, very much like the archetypal math lecture, with theorem, proof, and examples presented in sequence with sparse motivation. Notes can be taken, or, alternatively, one can follow the professor’s lecture with the textbook. The speed of the lectures was very quick, so many students and I preferred the second strategy, as copying notes detracted from following along the proofs. In either case, I recommend reading the material before going to class, because a lot of the motivation and discussion for certain topics that isn’t discussed in lecture. This way, one can stay motivated to learn the material, so it doesn’t seem like an endless, scattered list of propositions and lemmas. The class doesn’t seem to move at a breakneck speed, slower, but the two hour and a half lectures with another 50 minute section the means a lot of material is covered every week. Since the class is small, around 20-30 students, in-lecture questions should be asked, as the professor often may skip over steps that he deems sufficiently trivial.

Learning For and From Assignments

For: The midterm requires a pretty decent understanding of the techniques used in the proofs given in the book (which is also what a lot of the assignments were about). So carefully reading the book (as in any math class) is really important, but even more important is being able to glean out the major techniques used along the way. This includes, for example, the rigidity technique mentioned above, making use of analyticity to give some information “for free” about the function, and doing sometimes crude estimates (especially for Fourier Transforms and for thinking about zeros of the zeta function). Of course, as in any class, there are a couple of things you need to be a monkey for: practice contour integrals enough that you can do them in your sleep for they no doubt will appear on both the midterm and final. From: Ionescu doesn’t post solutions, and this goes for both homework and tests. For this reason, you’ve gotta keep them and within a week from getting them back go over everything. That means looking back at each problem and writing an outline of the key steps along the way (if you can’t then you should redo it in detail). If you still don’t understand any of the problems, write up everything you’re able to and take it into his office hours to have him go over with you what you’re missing.

External Resources

For Ionescu specifically: he teaches exactly from the book during lectures and will not expand unless pressured. However, if you do pressure him (e.g. after class or during office hours) he’s more than willing to give insights that go beyond the book and help you to understand. He also will go over extra problems with you, which is the key to understanding the material as (with most math classes) there are only a few problem-solving examples in the actual text. Complex Analysis and Solutions by Shakarchi is an awesome resource for getting extra practice, especially on the more fundamental first few chapters (and chapter 8 on Conformal Maps). Caveats: he starts with Conformal Maps, and skips some of the applications that we do while going into greater detail in others. Visualizations of functions in the complex plane were extremely useful for me, especially for the zeta, gamma, theta, and chi functions. They’re also crucial for getting a deep understanding of conformal maps (chapter 8). Wikipedia has all of the basic ones. You can also go to davidbau.com/conformal (just google david bau conformal maps and look around if this link doesn’t work). Further, there are Mathematica scripts available online for “doing it yourself” but they’re pretty slow. I never tried, but it probably wouldn’t be too hard to code up a complex function viewer yourself.

What Students Should Know About This Course For Purposes Of Course Selection

This is an essential course for any mathematics major, and is a good challenge for any other engineering and science majors. One should be comfortable with complex numbers and have taken at least two semesters of proof-based math before attending this course. It is not necessary to take Fourier Analysis (Analysis I) before taking complex, but it would be beneficial to because the style of the Princeton Analysis textbooks is quite unique. Complex Analysis, the second class in the Princeton Series in Analysis, also known as “real” Complex (as opposed the to applied complex analysis class), is widely regarded as the most difficult course in the undergraduate math curriculum. Thus, I would advise studying and getting familiar with some basics about complex analysis (i.e., chapters 1-3 of Stein and Shakarchi) during the break before. Also, one should expect over 20 hours of work per week outside of the classroom, so taking a five-course semester would be a very high workload, and de-loading on courses for the semester is a good way to ensure a deep understanding of the material.