Course: MAT330
Instructor: Aizenman
S 2015

Description of Course Goals and Curriculum

The theory of functions of one complex variable, covering power series expansions, residues, contour integration, and conformal mapping. Although the theory will be given adequate treatment, the emphasis of this course is the use of complex analysis as a tool for solving problems. The main goal of the course is to familiarize students with single-variable complex functions, and in particular, how extending functions to the complex plane yields new ways to work with functions on the reals. The course focuses on teaching students multiple strategies for solving certain mathematical problems that would otherwise have been very difficult or impossible.  A special emphasis is placed on creativity in problem-solving and proofs. Another point of emphasis is the application of complex analysis to other fields, such as, and especially, physics. The course can be roughly divided into two parts. The first half of the course is almost completely focused on developing the central theorems of complex analysis. It begins with the extension of real functions into the complex plane. It then expands to complex generalizations of topics learned in other math courses, such as Taylor series and the Fourier transform. However, initial familiarity with these topics, while advantageous, is not necessary, as the definitions are self-contained within the scope of the course. The second half of the course involves application of complex analysis to problems in other areas of study. In particular, a relatively large amount of time is spent on developing techniques for calculating integrals on the real numbers in a simple manner. No prior knowledge is assumed beyond the listed prerequisites. Knowledge of linear algebra, while officially a prerequisite, is mostly irrelevant within the context of the course.

Learning From Classroom Instruction

As with any mathematical course, complex analysis is best learned through practice. With this in mind, the professor briefly defines the relevant concepts before working through examples, usually distinct from the ones presented in the textbook. The lecture proceeds in a very organic manner, with a theme in mind but without any lecture notes. Therefore, taking adequate notes can prove to be very helpful when solving homework problems or studying for the exams. Because the lectures are not based on the textbook, the readings complement help to clarify any proofs or definitions not extensively covered in lecture. The readings are fairly concise and often include fewer examples than desired. This makes the examples in lecture serve as useful aids. As the professor advocates creative problem-solving, he often also provides alternative proofs to theorems. Comparing these to the methods shown in the textbook can be an effective learning tool. Being able to think of alternative proofs independently can really only be learned through experience -- therefore, while it is helpful to attempt multiple proofs, there is no reason to expend too much effort on a single case.

Learning For and From Assignments

Homework assignments are the primary way for students to become comfortable with the material. This course introduces complex analysis mainly as a tool for solving problems. Therefore, through the homework, students should focus on improving their problem-solving abilities within the context of complex analysis. The assignments consist of a combination of textbook and supplementary problems.  The supplementary problems are often directly relevant to examples or theorems covered in lecture. For the textbook problems, it is useful to read the relevant sections and examples, as they will usually outline the best methods for solving those in the textbook. An important part of problem-solving is being able to identify classes of problems which require different strategies. The exam problems are also similar to homework problems, in that the goal of the homework and exams is to make sure that students know how to apply what they have learned. None of the exam questions are inherently challenging, but they require students to understand conceptual subtleties and to recognize which strategies and methods are needed for specific problems. Homework assignments and exams from previous semesters serve as the best study aid, as the exams do not have much variation across semesters. The exams cover essentially the entire breadth of the material, so there is no sense in which a student should study specifically for the exam; instead, it suffices that the student actually understand how to use the methods learned and how to derive the most important results. Working out solved problems analogous to those on the problem sets is therefore the best method to study for exams.

External Resources

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

This course is a good introduction for anyone who is interested in complex analysis for practical purposes, as it discusses mathematical theory but does focus exclusively on it. The tools learned in the class are helpful for building intuition for computing certain types of integrals in a simple manner. They are especially applicable to physics. Complex analysis showcases the elegance of mathematics and the course can be rewarding, both for theory and for application.
Complex Analysis with Applications

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