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.).


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.
Wednesday, 4/22 Homework 6.
Wednesday, 5/6 Homework 7.

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.

Takehome Final Exam

The takehome midterm exam is here. It is due Wednesday May 13 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 and having admitting convergent Taylor expansions 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)
Mar. 16-20 No lecture (spring break).
Mar. 23 Lecture 24: 1-chains and 2-chains. Homologous cycles, and cycles that are homologous to zero (in the topological sense). The notion of being homologous to zero in the winding number sense, and a beginning of a proof of equivalence. Chapter 4 Section 4.1, 4.3 (p. 137-138, 141). These sections cover the definition of a cycle being homologous to zero in the winding number sense.
Mar. 25 Lecture 25: Proof of equivalence of the two notions of being homologous to zero (in the topological sense vs. in the winding number sense), part 1. For general topological spaces, the notion of a cycle being homologous to zero coincides with the definition we gave in class for a region in C (namely, that the cycle bounds a 2-chain). This definition is omitted in Ahlfors in favor of a computationally simpler definition (which we called "being homologous to zero in the winding number sense"); however note this latter definition only applies to regions of C (and not to arbitrary topological spaces).
Mar. 27 Lecture 26: Proof of equivalence of the two notions of being homologous to zero (in the topological sense vs. in the winding number sense), part 2. The general (homological) version of Cauchy's theorem and its proof. Simply connected regions (multiple variants of the definition given, and a theorem was stated that all definitions are equivalent). Chapter 4 Section 4.2 (p. 138-141), 4.4-4.5 (p. 141-144) Section 4.4 states the general form of Cauchy's theorem and Section 4.5 gives a proof, which is a bit different from the one we gave in class, because we can appeal to the topological version of being homologous to zero. Section 4.2 discusses simply connected regions. However, note Ahlfors defines simply connected regions to be any region "without holes", and showed that this is equivalent to any cycle in Omega being zero in the winding number sense, but does not discuss the topological notion of nullhomologous or the original definition of simply connected (that any loop is homotopic rel endpoints to a constant loop).
Mar. 30 Lecture 27: More about simply connected domains (and the various equivalent definitions). Proof that a region is simply connected (in either homological or winding number sense, that every cycle in the region is nullhomologous) if and only if it "has no holes" (meaning that its complement consists only of unbounded components, hence is connected in (C union infinity).) The notion of a "surrounding cycle" for a "hole" or bounded component A of (C minus Omega), a cycle in Omega that winds once around any point in A but zero times around other points in (C minus Omega). Existence of surrounding cycles. Some further consequences of a region Omega being simply connected (corollaries of Cauchy's theorem): its possible to define a single-valued branch of log f or nth root of f for any function on Omega without zeroes. Chapter 4 Section 4.2 (p. 138-141), 4.4 (p. 141-142), 4.7 (p. 146-147) Section 4.2 proves the equivalence of the two versions of the simply connected definition that appear in Ahlfors. (the proof given in class was slightly different). Note that Ahlfors does not use the term "surrounding cycle", but the construction of surrounding cycles is given in Section 4.7, by adapting an earlier construction given in the proof of Section 4.2's Theorem 14. Section 4.4 discusses applications of Cauchy's theorem to defining branches of log and nth roots on simply connected regions.
Apr. 1 Lecture 28: The notion of a homological basis of cycles, and the first homology of a region in C. For an arbitrary region, a collection of surrounding cycles around each of its holes (the bounded components of its complement) give a homology basis. The residue theorem. Chapter 4, Section 4.7 (p. 146 -147), Section 5.1 (p. 148-152) Section 4.7 discusses the notion of a "multiply-connected region" (meaning a region that's not simply connected) and a homology basis for it). The residue theorem is discussed in Section 5.1.
Apr. 3 Lecture 29: Recap of the residue theorem and techniques for computing residues. Conceptual applications: (1) the general version of the Cauchy integral formula, (2) the argument principle, a convenient formula for the number of poles and zeroes of a meromorphic function in a region (3) Rouche's theorem (an immediate corollary of the argument principle): which gives criteria for which a pair of functions have same number of zeroes in a region. An example of Rouche's theorem in practice. Chapter 4, Section 5.1-5.2 (p. 148-154).
Apr. 6 Lecture 30: Applications of the residue theorem to computing definite integrals 1. Chapter 4, Section 5.3 (p. 154-161).
Apr. 8 Lecture 31: Applications of the residue theorem to computing definite integrals 2. Cauchy principle value integrals. Chapter 4, Section 5.3 (p. 154-161).
Apr. 10 Lecture 32: Applications of the residue theorem to computing definite integrals 3. Sequences and series of complex analytic functions. Chapter 4, Section 5.3 (p. 154-161), Chapter 5 Section 1.1 (p. 175-179).
Apr. 13 Lecture 33: Weierstrass's theorem that a sequence of complex analytic functions which converges uniformly on compact subsets has a complex analytic limit. Hurwitz's theorem that if every term in the sequence is never zero, then the limit is either identically zero or never zero. Taylor and Laurent series. Chapter 5 Section 1.1-1.3 (p. 175-186).
Apr. 15 Lecture 34: Any complex analytic function on an annulus has a convergent Laurent expansion in that annulus. Integral representations for various terms in the Laurent expansion. Chapter 5 Section 1.3 (p. 184-186).
Apr. 17 Lecture 35: Normal families of functions and the Arzela-Ascoli theorem. Chapter 5 Section 5.1-5.3 (p. 219-223).
Apr. 20 Lecture 36: Normal families are relatively compact (or "pre-compact") subsets with respect to a metric/topology on function space. A simplified criterion for normality when the functions in the family are complex analytic: A family of complex analytic functions is normal if and only if it is locally bounded or equivalently uniformly bounded on compact subsets. Chapter 5 Section 5.1-5.4 (p. 219-225).
Apr. 22 Lecture 37: Biholomorphisms between regions of (the extended) complex plane. The classification of all biholomorphisms of the extended complex plane as Mobius transformations (also known as fractional linear transformations). The cross ratio of a 4-tuple in the extended plane. Chapter 3 Section 3.1-3.2 (p. 76-80) Section 3.1 introduces fractional linear transformations (which are often simply called linear transformations in the book). The proof that these are all biholomorphisms seems implicit but not explicitly mentioned (compare Chapter 4 Section 3.2 exercise 4 on p. 130, which asks the reader to show the key first step in the classification, that all biholomorphisms of C union infinity are rational functions).
Apr. 24 Lecture 38: Fractional linear transformations send generalized circles (which are by definition either a circle in C or a line in C union the point at infinity) to generalized circles. The subgroup of fractional linear tranformations that preserve the unit disk. The classification result that any biholomorphism of the unit disk is one of these (unit-disk-preserving) Mobius transformations. Chapter 3 Section 3.2 (p. 78-80). The classification of the biholomorphisms of the unit disk, a corollary of Schwarz's Lemma, appears as an exercise in Ahlfors (see Chapter 4 Section 3.4 exercise 5 (p. 136))
Apr. 27 Lecture 39: Explicit biholomorphisms between more general simply connected regions in the complex plane. Examples of biholomorphisms induced by the other elementary functions (like exp, and log, or real powers, the latter two of which are only defined on a subregion of the plane after choosing a branch). Biholomorphisms between the unit ball and the upper half plane (and vice versa). Examples of more complicated explicit biholomorphisms as a composition of various elementary biholomorphisms. Chapter 3 Section 4.1-4.2 (p. 89-97).
Apr. 29 Lecture 40: The Riemann mapping theorem part 1. Chapter 6 Section 1.1 (p. 229-232).
May 1 Lecture 41: The Riemann mapping theorem part 2. Chapter 6 Section 1.1 (p. 229-232).