General Information
This is the second half of the two-course qualifying sequence for Analysis. For graduate students in the mathematics department, this course is intended to be the part of the preparation for the qualifying exam in Complex Analysis; see the graduate student handbook for more information concerning the qualifying exams.
Complex Analysis is an umbrella term covering three sets of ideas, which trace to the works of Cauchy, Weierstrass, and Riemann respectively.
- Cauchy focused on understanding the differential and integral calculus of complex valued functions of one complex variable. It turns out that complex differentiability is a much more rigid phenomenon than its real counterpart, and this leads to analytic functions (with isolated singularities) being central objects of study. For the integral calculus, the fundamental ideas are captured in the Cauchy integral theorem and the Cauchy integral formula, applications of which will occupy a significant portion of this course.
(It is worth noting that a lot of what goes on in this set of ideas generalize beyond complex analysis. The domains of applicability of the Cauchy integral approaches are naturally tied to questions of topology. And complex differentiability and consequences of the integral formula are revived also in the study of elliptic partial differential equations, in the context of real variable differential calculus.) - Weierstrass took the foundations of from Cauchy's work, which asserted that complex differentiable functions have power series representations, and ran with it. From Weierstrass we acquired a branch of complex analysis whose signature is the manipulation of power series (including their sums, products, factorizations).
- Riemann regarded the complex-valued functions of a complex variable as (partial) function from the complex plane to itself, and focused their geometric aspects. A fundamental idea is that of a conformal mapping, and among the more well-known results in this direction one finds the Riemann Mapping Theorem and the Uniformization Theorem.
As a first introduction to complex analysis, this course will focus mostly on the strand that is associated to Cauchy above, but we will try to include some discussion of topics more in the Weierstrass and Riemann flavors.
Textbook and Topics
I will primarily follow Part 2A: Basic Complex Analysis from Barry Simon's A Comprehensive Course in Analysis. As the book is truly comprehensive (there's even a second volume, Part 2B: Advanced Complex Analysis), it is impossible to cover everything in it. My tentative plan is:
- Pre-requisites Students are assumed to be familiar with basic notions in sections 1.1-1.4 of the textbook. Material in sections 1.5-1.9 will be introduced as necessary.
- Base The main part of the course will focus on the entirety of Chapters 2 and 3 of the textbook, and sections 5.1 and the three-line theorem from section 5.2.
- Extensions A selection of the topics from the following list will be included.
- The "Ultimate Cauchy Integral Theorem" from Chapter 4.
- Additional applications of the integral theorem from Chapter 5.
- Spaces of Analytic Functions from Chapter 6.
- Geometric points of view from Chapters 7 and 8.
- Factorization (product formulae) of Analytic Functions from Chapter 9.
A topic that we will definitely not touch upon, but has an important place in modern complex analysis, is Nevanlinna's theory about (roughly speaking) counting the number of solutions to $f(z) = a$ and studying how this depends on $a$. Some of its precursors will appear in this course, but not the very important Great and Little Picard Theorems.
Complex Analysis is also used to study asymptotic behaviors of functions. This topic of great importance to the applied mathematician includes Paley-Wiener theorems, stationary phase and steepest descent arguments, and the WKB approximation. We will also not touch upon these subjects, even though the course should provide sufficient background for students to pursue these ideas on their own.
For students wishing alternative references, a book I have used in the past is Lars Ahlfors' Complex Analysis. With regards to the 1966 edition, the Base material are found in chapters 2, 4, and parts of 3 and 5. The Extensions can mostly be found in chapters 3, 5, and 6. For those preferring John B. Conway's Functions of One Complex Variable I, in the second edition the Base material are in chapters 3, 4, 5, and 6. The Extensions are can mostly be found in chapters 7, 8, and 9.
Practical information
Instructor
Professor Willie W. Wong
Office: D303 Wells Hall
E-mail: wongwwy@math
Office hours: by appointment
Homework assignments
TBA
- HW1 is due TBD
Exams
Grading
Each exam will account for X% of your final grade; each homework assignment will make up Y%.
Collaboration is encouraged for the homework assignments. Collaboration is not allowed on the in-class exams. The general MSU rules concerning academic integrity applies.