# Math 520: Complex Analysis, Spring 2020

## Monday, Wednesday, Friday, 1:00-1:50pm in KAP 134

### Teaching Staff

 Instructor Sheel Ganatra TA Zachary Wickham Office KAP 266D Office KAP 248A e-mail sheel (dot) ganatra (at) usc (dot) edu (this is the best way to reach me) e-mail zwickham (at) usc (dot) edu Office Hours This week: Wed 2-3 and by appointment. Office Hours TBD

### Course Description and Prerequisites

Math 520 is a graduate-level introduction to complex analysis, which is broadly the study of complex differentiable functions of one (complex) variable. Remarkably, the condition that a function be complex differentiable is an incredibly strong and rigid one: it is automatically infinitely differentiable and analytic (and hence is often just called a complex analytic function), and has a quite rigid behavior. This behavior can and will be studied from a variety of viewpoints, and we will explore consequences and applications, examples, and the (intimately related) theory of harmonic functions.

Prequesite-wise, we will assume some familiarity with real analysis at the level of Math 425, and comfort with writing proofs at an (introductory) graduate level. Concretely, we will essentially assume (but sometimes briefly review) basic facts about the real (and to some extent complex) numbers, limits, continuity, calculus (differentiability/integration of real functions), metric topology (open, closed, and compact sets, etc.).

### Announcements

• 1/25/2019:From Friday 1/17/2019 onwards, we will be meeting in KAP 134 (at the same time as before, 1pm).
• 1/13/2019: Welcome to Math 520! Please familiarize yourself with the course website/syllabus and get a copy of the textbook, if you haven't already.

### Textbook and topics

The official course text is: Complex Analysis, Lars Ahlfors, McGraw-Hill (3rd edition). We may deviate from this book; if so additional references will be posted as needed.

Some topics to be covered (subject to change) include:

• Complex numbers
• Complex differentiable (or holomorphic) functions and (later) meromorphic functions
• Complex integration: Cauchy's theorem, Cauchy's integral formula, residues and residue integrals
• Liouville's theorem and (using it) the fundamental theorem of algebra
• Power series. Complex differentiable functions are analytic, hence often just called complex analytic.
• Conformal mappings and the Riemann mapping theorem
• Harmonic functions and the Dirichlet problem
• Analytic continuation
• (time permitting) Special functions (zeta function, Gamma function, elliptic functions) and applications

A more detailed lecture plan (updated on an ongoing basis, after each lecture) will be posted below.

• 50% Homework assignments,
• 20% midterm,
• 30% final exam.

### Homework Assignments

Homeworks will be posted here on an ongoing basis (roughly a week or more before they are due) and will be due on the date listed. You can handwrite or LaTeX your solutions. You may work with others and consult references (including the course textbook), but the homework you turn in must be written by you independently, in your own language, and you must cite your sources and collaborators.

Note: Homework deadline extensions are possible upon arrangement with me (with our TA, CC'ed) up to 3 times a semester, and at most once per HW. If you decide to request an extension from me, there will be no grading penalty if the HW is submitted within a couple days -- I'll set a more precise deadline each request. Beyond this, late homework will not be accepted.

Due date Assignment
Friday 1/24 Homework 1.
Friday, 1/31 Homework 2.

### Lecture Plan

Lecture topics by day will be posted on an ongoing basis below. Future topics are tentative and will be adjusted as necessary.

Day Lecture topics References and remarks (in Ahflors unless stated) Notes
Jan. 13 Welcome and overview of class. Real and complex numbers (basic definitions). Chapter 1
Jan. 15 More on complex numbers: conjugate, norm. Real part and imaginary part give an identification C = R^2. Complex mutplication is multiplicative on norms and additive on arguments. Real linear maps from C = R^2 to C = R^2 (which are multiplication by a 2 x 2 matrix), and complex linear maps (which are multiplication by a complex scalar), and various characterizations of which real-linear maps are in fact complex linear. Chapter 1 Section 1 (p 1-12)
Jan. 17 A review of limits and differentiability of real (multivariable) functions, partial derivatives. Complex differentiability of a function f: C to C and its relationship to real differentiability of the same function ("Complex differentiability is real differentiability along with the total derivative matrix being a complex linear map"). The Cauchy-Riemann equations. Harmonic functions. Chapter 1 Section 2.1 (p. 12-15), Chapter 2 Section 1.1-1.3 (p. 21-29) Note that Ahlfors calls functions which have a well-defined complex derivative complex analytic whereas we will for now use complex differentiable or holomorphic, and only use the term analytic once we prove the (non-trivial) theorem that complex differentiable functions are in fact analytic (in the sense of being infinitely differentiable and equal to its Taylor expansion in a neighborhood of any point).
Jan. 20 No class (MLK day)
Jan. 22 The Cauchy-Riemann equations, harmonic functions, and conjugate harmonic functions (if they exist). Limits, sequences, and series (a review of concepts from real analysis). Same as last class, plus Chapter 2 section 2.1-2.2 (p. 33-35)
Jan. 24 Series and power series, convergence and absolute convergence. Abel's theorem on convergence of power series (in a real or complex variable). Chapter 2 Section 2.2-2.4 (p. 35-40)