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Complex analysis
Math 520, USC Fall 2025

Instructors.

Kyler Siegel (kyler.siegel@usc.edu)

Teaching assistants.

Yasin Uskuplu (uskuplu@usc.edu)

Time and location.

11am - 11:50am on Mondays, Wednesdays, and Fridays in 414 Kaprielian Hall (KAP). Note: the first class will exceptionally meet in KAP 265.

Office hours.

Kyler: typically Wednesdays and Fridays 1:30-2:30pm in KAP 400C (these may get modified by an announcement from time to time)
Yasin Uskuplu: Wednesdays 5-7pm over Zoom (meeting id 983 4250 5194)

About.

This course is an introduction to graduate level complex analysis. The main objects of study are functions of one complex variable which are holomorphic, i.e. complex differentiable. At first glance one might expect the whole theory to closely resemble ordinary calculus of a single real variable, but in fact holomorphic functions have many remarkable features not present in the real case. In the first part of the course we will develop many fundamental ideas in complex analysis, such as holomorphic and meromorphic functions, Cauchy's theorem, integration via residues, entire functions, the Fourier transform, the gamma and zeta functions, and so on. We will then move on to more advanced topics such as the prime number theorem, conformal mappings, elliptic functions, Riemann surfaces, and more. The topics we get to in the latter part of the course will depend partly on the pace and interests of the audience.

Prerequisites.

We will assume familiarity with real analysis. Some knowledge of topology will also be helpful.

Topics.

Fundamentals:

complex numbers, the complex plane, and point set topology
holomorphic functions, power series
integrals along curves and Cauchy's theorem
mermorphic functions, residues, and the singularity trichotomy
definite integrals and contour integration, the argument principle
the Fourier transform
entire functions, infinite product representations
the gamma and zeta functions

Possible advanced topics:

the prime number theorem
conformal mappings
elliptic functions
theta functions and sums of squares
Riemann surfaces.

Textbooks and references.

Complex analysis - Stein and Shakarchi [SS]
Riemann surfaces - Donaldson [D]

Problem sets.

There will be weekly problem sets. These will be listed on this website (see the schedule below) and will be handed in via Gradescope. Late homeworks will not be accepted without prior approval from the instructor.

Much of the course material will be developed in the problem sets. It is very important to do all of the problem sets to the best of your ability and to challenge yourself to solve the problems on your own, as this is the most effective way to absorb the material. We expect students to devote a significant amount of time to the problem sets.

While working on the problem sets, you are allowed to collaborate with your peers and to consult textbooks or other references. However, you must write down attributions for any peer, textbook, website etc from which you took any significant ideas. Moreover, you must attempt all problems on your own and your submitted solutions must be written out originally and individually. Submissions which are copied or suspiciously similar are subject to being rejected and potential disciplinary action.

Final presentations.

In lieu of a midterm, we have opted to have final student presentations. More details on the structure and potential topics will be given after spring break.

Grading scheme.

TBA

Tentative schedule

Note: this schedule (along with the above information) is tentative and will be continuously updated to adapt to the pace of the course. Please check back regularly for updates and problem set assignments.

$\#$	Date	Material	References	Problem set
1	Monday 1/13/25	Basics algebraic and topology properties of the complex numbers. Definition of holomorphic functions.	[SS] Chapter 1 §1,§2
2	Wednesday 1/15/25	The Cauchy-Riemann equations and holomorphicity. Relationship with real differentiability.	[SS] Chapter 1 §2.2	Pset 1 (due Thurs 1/23/25 at 11:59pm)
3	Friday 1/17/25	Recap on Cauchy-Riemann equations in terms of complex linearity of the Jacobian. Proof that the Cauchy-Riemann equations imply holomorphicity. Power series and their radii of convergence. Hadamard's formula.	[SS] Chapter 1 §2.2, §2.3
	Monday 1/20/25: MLK day - no class
4	Wednesday 1/22/25	Holomorphicity of power series. Piecewise smooth curves. Integrals along curves and their basic properties. Primitives.	[SS] Chapter 1 §2.3, §3	Pset 2 (due Friday 1/31/25 at 11:59pm)
5	Friday 1/24/25	Statement of Cauchy's theorem, and proof in the continuously differentiable case. Proof of Goursat's theorem.	[SS] Chapter 2 §1.
6	Monday 1/27/25	Cauchy's theorem in a disc and its consequences, and generalizations to "toy contours". First example of a definite integral computation using contour integration.	[SS] Chapter 2 §2,§3.
7	Wednesday 1/29/25	More examples of definite integrals by contour integration. The standard Gaussian integral. Cauchy's integral formula and its proof.	[SS] Chapter 2 §3,§4.
8	Friday 1/31/25	More on Cauchy's integral formula and its extension to higher derivatives. Holomorphic functions are infinitely differentiable (in particular Green's theorem holds). Cauchy's derivative bounds. Liouville's theorem and the fundamental theorem of algebra.	[SS] Chapter 2 §4.	Pset 3 (due Weds 2/12/25 at 11:59pm)
9	Monday 2/3/25	Proof that holomorphic functions have convergent power series. Implications for radii of convergence. Vanishing on an open set implies vanishing everywhere. Analytic continuation. Morera's theorem.	[SS] Chapter 2 §4,§5.1.
10	Wednesday 2/5/25	A uniform limit of holomorphic functions is holomorphic. Holomorphic functions defined by integrating out a parameter. The symmetry principle and Schwarz reflection principle.	[SS] Chapter 2 §5.2, §5.3, §5.4.
11	Friday 2/7/25	Approximating holomorphic functions on compact sets by polynomial and rational functions. Pole and zero orders of holomorphic functions. Informal statement of the residue formula.	[SS] Chapter 2 §5.5, Chapter 3 §1,§2.
12	Monday 2/10/25: asynchronous class	More on the singularity trichotomy. Proof of the residue formula. Examples of computing real definite integrals using the residue formula. The removable singularity theorem.	[SS] Chapter 3 §1,§2.
13	Wednesday 2/12/25	Recap from last time. Characterization of poles in terms of $\|f(z)\|$ . Statement and proof of the Casorati-Weierstrass theorem on essential singularities. Statement of the great Picard theorem. Meromorphic functions. The extended complex plane.	[SS] Chapter 3 §1,§2,§3.	Pset 4 (due Fri 2/21/25 at 11:59pm)
14	Friday 2/14/25	Stereographic projection and the Riemann sphere. Meromorphic functions on the extended complex plane are rational functions. The argument principle.	[SS] Chapter 3 §3,§4.
	Monday 2/17/25: President's Day: no class
15	Wednesday 2/19/25	Review of the argument principle. Rouché's theorem, the open mapping theorem, and the maximum modulus principle. Simply connected domains and simply connected version of Cauchy's theorem.	[SS] Chapter 3 §4,§5.
16	Friday 2/21/25	Harmonic functions and the mean value property. More on the complex logarithm, its branches, and the function $z^\lambda$ for $\lambda \in \mathbb{C}$ . Brief introduction to the Fourier transform.	[SS] Chapter 3 §6,§7.	Pset 5 (due Fri 2/28/25 at 11:59pm)
17	Monday 2/24/25	The Fourier transform of functions in class $F_a$ . Proof that such functions have Fourier transforms with exponential decay. Proof of the Fourier inversion formula.	[SS] Chapter 4 §1,§2.
18	Wednesday 2/26/25	Proof of the Poisson summation formula. Theta function example. Statement that functions with exponentially decaying Fourier transform extend holomorphically to a strip. Weierstrass's existence theorem on entire functions with prescribed zeros and first observations.	[SS] Chapter 4 §1,§2, Chapter 5 §3,§4.	Pset 6 (due Fri 3/7/25 at 11:59pm)
19	Friday 2/28/25	Preliminaries on convergence of infinite products. Canonical factors and the proof of Weierstrass' existence theorem.	[SS] Chapter 5 §3,§4
20	Monday 3/3/25	Order of growth of entire functions. Theorem relating the zeros of an entire function with its order of growth. Jensen's formula.	[SS] Chapter 5 §1,§2.
21	Wednesday 3/5/25	Hadamard's theorem and the example of $\sin(z)$ . Preliminaries on biholomorphisms of open subsets of $\mathbb{C}$ . Proof that the inverse of an invertible holomorphic map is holomorphic.	[SS] Chapter 5 §5, Chapter 8 §1.
	Friday 3/7/25 - no class			Pset 7 (due Fri 3/14/25 at 11:59pm)
22	Monday 3/10/25	Examples of biholomorphisms between open subsets. The upper half space is biholomorphic to the unit disc. The (partial) extension to the boundary.	[SS] Chapter 8 §1.
23	Wednesday 3/12/25	More examples of explicit biholomorphisms between various domains in $\mathbb{C}$ . Solving the Dirichlet problem for Laplace's equation in the unit disc, and its extensions to other domains using biholomorphisms. Automorphisms of domains.	[SS] Chapter 8 §1.
24	Friday 3/14/25	Computing the automorphism groups of the unit disk and the upper half plane. The Schwarz lemma. M\"obius transformations and their properties. The automorphism group of the extended complex plane.	[SS] Chapter 8 §1, §2.
	Monday 3/17/25: spring break - no class
	Wednesday 3/19/25: spring break - no class
	Friday 3/21/25: spring break - no class
25	Monday 3/24/25	More on the automorphism groups of $\mathbb{D}, \mathbb{H}, \widehat{\mathbb{C}},\mathbb{C}$ and their topology. Overview sketch proof of the Riemann mapping theorem modulo various gaps.	[SS] Chapter 8 §3.
26	Wednesday 3/26/25	Recapping the overview proof of the Riemann mapping theorem. Filling in all of the gaps except for the proof of a convergent subsequence.	[SS] Chapter 8 §3.	Pset 8 (due Wednesday 4/2/25 at 11:59pm)
27	Friday 3/28/25	Montel's theorem and the remaining gap in the proof of the Riemann mapping theorem. The Schwarz-Christoffel formula for biholomorphisms between the upper half plane and a polygon. Winding number of a point around a contour and geometric significance of the argument principle. Justification of the formula in terms of arguments and the examples of $\sin^{-1}(z)$ .	[SS] Chapter 8 §3,§4.
28	Monday 3/31/25	Introduction to Riemann surfaces. Definitions of Riemann surfaces and smooth surfaces. Atlas and chart terminology. Crash course in topology (topological spaces, continuous maps, homeomorphisms). Every compact Riemann surface is homeomorphic (resp. diffeomorphic) to a genus $g$ topological (resp. smooth) surface. The Riemann sphere and its atlas with two charts.	[D] §3.1,§3.2.1.
29	Wednesday 4/2/25	Recap on Riemann surfaces and the example of the Riemann sphere. Holomorphic maps between Riemann surfaces. Pedantic distinction between equal, equivalent, and biholomorphic Riemann surfaces. Definition of complex manifolds. Complex projective space as a complex manifold and its natural atlas of charts, and the picture $\mathbb{CP}^n = \mathbb{C}^n \cup \mathbb{CP}^{n-1}$ .	[D] §3.1,§3.2.1.	Pset 9 (due Wednesday 4/9/25 at 11:59pm)
30	Friday 4/4/25	Introduction to algebraic curves in $\mathbb{CP}^2$ . Complex analytic proof of the implicit function theorem.	[D] §1.1,§3.2.2.
31	Monday 4/7/25	More on algebraic curves in $\mathbb{C}^2$ and using the implicit function theorem to construct charts. Homogeneous polynomials and their vanishing loci in $\mathbb{CP}^2$ . Euler's identity and the Riemann surface structure on nonsingular algebraic curves in $\mathbb{CP}^2$ . Some simple examples. The degree genus formula.	[D] §3.2.2.
32	Wednesday 4/9/25	More in projective algebraic curves. Construction Riemann surfaces by quotients of group actions, and the relevant conditions needed. Free actions and discrete subgroups of $PSL(2,\mathbb{R})$ . Congruence subgroups of $PSL(2,\mathbb{Z})$ .	[D] §3.2.3.
33	Friday 4/11/25	Brief review of our constructions of Riemann surfaces so far. Statement and proof of the inverse function theorem. Proof that holomorphic maps between Riemann surfaces are locally of the form $z \mapsto z^k$ . Definition of the local degree of a map at a point in the domain. Definition of the (global) degree of a holomorphic map and proof that it well-defined.	[D] §4.1.
34	Monday 4/14/25	Recap on the local structure of holomorphic maps between Riemann surfaces and local and global degrees. The role of properness in the noncompact case. Crash review on fundamental groups. Holomorphic maps give transitive permutation representations of the target Riemann surface minus the discriminant locus.	[D] §4.2.1, §4.2.2.
35	Wednesday 4/16/25	Statement and proof sketch of the Riemann existence theorem. Meromorphic functions on the Riemann sphere, i.e. rational functions. Some examples.	[D] §4.2.1, §4.2.2.
36	Friday 4/18/25	More on fundamental groups of punctured surfaces and transitive permutation representations. The Riemann-Hurwitz formula. Proof sketch using the Euler characteristic. Various examples, including degree two covers of the Riemann sphere, and the construction of a genus two Riemann surface.	[D] §7.2.1.	Pset 10 (due Sunday 4/25/25 at 11:59pm)
37	Monday 4/21/25	Review of Riemann-Hurwitz. Examples coming form projections of algebraic curves. The Riemann surface associated to a germ of a holomorphic function.	[D] §4.2.4.
38	Wednesday 4/23/25	More on the construction of the Riemann surface associated to a germ of a holomorphic function, and various examples. Ordinary differential equations and the holonomy near regular singular points.	[D] §4.2.4, §1.2.
39	Friday 4/25/25	Introduction to uniformization. Statement of the Riemann--Roch theorem and using it to produce nonconstant meromorphic functions on Riemann surfaces, and in particular uniqueness of the Riemann sphere. The moduli space of genus $g$ Riemann surfaces and its dimension. The moduli space of genus one Riemann surfaces in two ways: as degree two covers of $\mathbb{CP}^1$ , and as $\mathbb{H} / \operatorname{PSL(2,\mathbb{Z})}$ . Fundamental domain for the action of $\operatorname{PSL}(2,\mathbb{Z})$ on $\mathbb{H}$ .
40	Monday 4/28/25 Note: we start at 10:45am and meet in KAP 166	Final presentations: (1) the gamma function (Nicholas) (2) the zeta function (Zihan)
41	Wednesday 4/30/25 Note: we start at 10:45am and meet in KAP 150	Final presentations: (3) zeros of the zeta function (Mengxi) (4) the prime number theorem (Scott)
42	Friday 5/2/25 Note: we start at 10:45am and meet in KAP 150	Final presentations: (5) linear differential equations (Ruoyu) (6) theta functions (Ziheng)
42	Monday 5/5/25 Note: we start at 10:45am and meet in KAP 134	Final presentations: (7) theta functions and sums of squares theorems (Yuanjia) (8) Applications of the residue theorem to evaluating integrals and infinite sums (Metehan)
Final exam: Monday 5/12/25 from 1-4pm in KAP 414.

Image sources: here.

Complex analysis Math 520, USC Fall 2025