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 Mon 3:15-4:15pm, Wed 10-11am, Thurs 3-4pm, and available for appointment other times (all by Zoom). Office Hours (Every week) Tues 2-4, Wed 2-3.

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

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:

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

Grading System

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.
Monday, 2/10 Homework 3.
Tuesday, 3/3 Homework 4.
Monday, 3/9 Homework 5.

Takehome Midterm Exam

The takehome midterm exam is here. It is due Thursday April 9 at 5 pm by e-mail to the instructor and TA.

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 Lecture 1: Welcome and overview of class. Real and complex numbers (basic definitions). Chapter 1
Jan. 15 Lecture 2: 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 Lecture 3: 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 Lecture 4: 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 Lecture 5: 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)
Jan. 27 Lecture 6: The Weierstrass M-test and the proof of Abel's theorem. Chapter 2 Section 2.2-2.4 (p. 35-40)
Jan. 29 Lecture 7: Exponential and trignometric functions (defined via their series and/or differential equations). The periods of the exponential function are integer multiples of 2 pi i. Chapter 2 Section 3.1-3.3 (p. 42-46)
Jan. 31 Lecture 8: Logarthims and exponentiations of complex numbers by other complex numbers (both functions which in general are multi-valued, unless one chooses a branch i.e., specifies some sort of domain over which the logarithm is both single-valued and continuous/differentiable. A review of topology: metric spaces, open and closed sets, continuous functions, uniformly continuous functions. Chapter 2 Section 3.4 (p. 46-49) Chapter 3 Section 1.1-1.2 (p. 50-54), 1.5 (p. 63-66), Chapter 3 Section 2.2 (p. 69-72)
Feb. 3 Lecture 9: More topology review: connected and locally connected subsets of metric spaces, complete metric spaces, and compactness. The Bolzano-Weierstrass theorem and Heine-Borel theorems. The intermediate value theorem. Any continuous function on a compact set is uniformly continuous, and --- if it is real-valued --- attains a max and a min. Chapter 3 Section 1.1-1.5 (p. 50-66).
Feb. 5 Lecture 10: Complex differentiable maps (when their derivative isn't zero) as conformal maps (maps that preserve angles between tangent vectors but scale and rotate a variable amount). Conformal equivalences/biholomorphisms/complex diffeomorphisms of regions in c. The Riemann mapping theorem (statement). Integration of parametrized curves. Chapter 3 Section 2 (p. 67-75)
Feb. 7 Lecture 11: (Parametrized) curves in C, also known as arcs (which are assumed piecewise continuous or differentiable dependending, but for some purposes need only be rectifiable). Operations on curves: Concatenation, reversal. Line integrals (integrals of functions along curves). Chapter 4 Section 1.1-1.3 (p. 101-109)
Feb. 10 Lecture 12: More line integrals. A study of when line integrals only depend on the endpoints of the curve. The line integrals of f(z) dz only depends on the endpoints if f is the complex derivative of a complex differentiable function defined on some region containing the curve. Chapter 4 Section 1.1-1.3 (p. 101-109)
Feb. 12 Lecture 13: Cauchy's theorem for a rectangle. Chapter 4 Section 1.4 (p. 109-112)
Feb. 14 Lecture 14: Cauchy's theorem for a disk. Winding numbers of closed curves (about points not in their image). The Cauchy integral formula. Chapter 4 Section 1.5, Chapter 4 Section 2.1-2.2 (p. 112-120).
Feb. 17-21 No class.
Feb. 24 Lecture 15: The Cauchy integral formula (or integral representation formula) for the higher derivatives of a complex differentiable function (and in particular, the proof that complex differentiable functions have higher derivatives). Chapter 4 Section 2.3 (p. 120-123)
Feb. 26 Lecture 16: Applications of the Cauchy integral formula: Morera's theorem, the Cauchy estimate, Liouville's theorem, the fundamental theorem of algebra. The Cauchy integral formula and Cauchy's theorem remain valid in the presence of removable singularities, and hence removal singularities can in fact be removed. Chapter 4 Section 2.3 (p. 120-123)
Feb. 28 Lecture 17: Taylor's theorem for complex differentiable functions. Complex differentiable functions are complex analytic: any complex differentiable function has a convergent Taylor series expansion at any point in its domain. Chapter 4 Section 3.1 (p. 124-126), Chapter 5 Section 1.2 (p. 179-180 particularly)
Mar. 2 Lecture 18: A complex analytic functions has no infinite order zeroes unless it is identically zero, The necessarily finite-order zeroes of a not-identically-zero function are isolated. In particular, if two analytic functions agree on some subset with accumulation points, they agree everywhere. Isolated singularties of functions. Poles (a particular type of isolated singularities). Chapter 4 Section 3.2 (p. 126-130).
Mar. 4 Lecture 19: Classification of isolated singularities as removable, poles, or essential singularities A limit characterization of each of these singularities. A function comes arbitrarily near any value in any neighborhood of an essential singularity. Chapter 4 Section 3.2 (p. 126-130).
Mar. 6 Lecture 20: More essential singularities. Open mappings and the open mapping theorem. An immediate corollary, the maximum principle. A quantitative refinement of the open mapping theorem, which characterizes the number of solutions to f(z) = w for w near a point w_0 and z near a point Chapter 4 Section 3.2-3.4 (p. 126-137).
Mar. 9 Lecture 21: Conclusion of the proof of the open mapping theorem (and its quantitative refinement). A winding number characterization of the number of zeroes of a function in a disc. Chapter 4 Section 3.3-3.4 (p. 130-137).
Mar. 11 Lecture 22: Another proof of the maximum principle. The Schwarz Lemma. Chapter 4 Section 3.4 (p. 133-137).
Mar. 13 Lecture 23: Towards the general formulation of Cauchy's theorem: (1-)chains, which are particular formal sums of parametrized curves up to equivalence, and cycles (which are those chains that can be represented by sums of closed curves). Two cycles are homologous if their difference is nullhomologous or homologous to zero (along with one heuristic and one precise definition of this latter term). Chapter 4 Section 4.1, 4.3 (p. 137-138, 141)