# Complex analysis (Math 1260)

Undergraduate course, *Brown University*, 2021

## Course summary

This is an undergraduate course about the complex numbers ℂ and the study of functions ℂ → ℂ that are complex differentiable (these functions are called “holomorphic”). This is a rich and beautiful subject with connections to many other areas of math and beyond. We will focus on the theoretical aspects of complex analysis (the “what” and the “why”), although we will also discuss computations and applications (the “how” and the “who cares”).

Some highlights/goals of this course:

- learn foundational results about holomorphic functions (e.g. Cauchy-Riemann equations, Cauchy integral formula, Riemann mapping theorem)
- learn applications of complex analysis to other areas (fundamental theorem of algebra, contour integration)
- strengthen your understanding of calculus/analysis by seeing it “in action”
- improve your ability to read/write proofs and critique arguments

## Textbook

Gamelin, *Complex Analysis*

We will mainly cover Parts 1 and 2, but may do some of Part 3.

## Topic schedule

- Week 1 (Lecture 1): Gamelin Chapter I, Section 1-2, 5
- Thurs (9/9). Similarities/Differences between ℝ and ℂ, arithmetic (algebraically and geometrically), complex exponential (notes)

- Week 2 (Lectures 2-3): Gamelin Chapter I, Sections 4,7,8; Chapeter II, Sections 1-3
- Tues (9/14). Complex exponential, derivatives, roots (notes)
- Thurs (9/16). holomorphic functions, CR equations, complex trig/logarithm functions (notes)
- Fri (9/17). HW1 due.

- Week 3 (Lectures 4-5): Gamelin Chapter 2, Sections 5; Chapter 3, Sections 1-3, Chapter 4, Sections 1-2
- Tues (9/21). Harmonic functions, complex integration
- Thurs (9/23). Harmonic conjugates, complex integration, Green’s theorem and Cauchy’s theorem
- Fri (9/24). HW2 due

- Week 4 (Lectures 6-7): Gamelin Chapter 4, Sections 3-6
- Tues (9/28). Cauchy’s theorem, Green’s theorem proof, antiderivatives
- Thurs (9/30). Cauchy integral formula, holomorphic implies smooth, Liouville and Morera’s theorems
- Fri (10/1). HW3 due

- Week 5 (Lectures 8-9): Gamelin Chapter 5, Sections 4,7,8
- Tues (10/5). Cauchy integral formula, computing with CIF, holomorphic implies analytic
- Thurs (10/7). residue calculus, fundamental theorem of algebra
- Fri (10/8). HW4 due

- Week 6 (Lectures 10, midterm): Gamelin Chapter 5, Section 8
- Tues (10/12). Analytic continuation, residue theorem computations
- Thurs (10/14). Take home midterm (no lecture)
- Fri (10/15). Midterm due on Gradescope by 5pm

- Week 7 (Lectures 11-12):
- Tues (10/19). analytic continuation, Gamma/zeta functions, more residue calculus
- Thurs (10/21). More on Gamma/zeta functions, summation with the residue theorem
- Fri (10/22). HW5 due

- Week 8 (Lectures 13-14): Ch 3, Section 5; Ch 9, Sections 1-2
- Tues (10/26). More on Gamma/Zeta functions, maximum modulus principle
- Thurs (10/28). Maximum principle for harmonic functions, disk automorphisms, Singularities
- Fri (10/29). HW6 due

- Week 9 (Lectures 15-16): Chapter 6, Sections 1-2; Chapter 9, Sections 1-2
- Tues (11/2). Singularities, Schwarz lemma, Mobius transformations, Laurent series decomposition
- Thurs (11/4). Laurent series decomposition, Singularities, Argument principle
- Fri (11/5). HW7 due

- Week 10 (Lectures 17-18): Ch 8, Sections 1-4
- Tues (11/9). Argument principle, Rouche’s theorem, non-isolated singularities
- Thurs (11/11). Open mapping theorem, Maximum modulus (revisited), inverse mappings
- Fri (11/12). HW8 due

- Week 11 (Lectures 19-20): Chapter 11, Sections 2, 5, 6
- Tues (11/16). Time to work on your project (no class)
- Thurs (11/18). SL(2,ℤ) and the upper half plane, Riemann mapping theorem
- Fri (11/19). HW9 due (last homework assignment)

- Week 12 (Lecture 21):
- Tues (11/23). More SL(2,ℤ), Riemann mapping theorem
- Thurs (11/25). No class (Thanksgiving holiday)

- Week 13: Final projects
- Quinn and Seth: Picard theorems
- Andrew and Sejin: Weierstrass product theorem
- Gareth and Robert: Mobius transformations
- Sean and Mark: the Mandelbrot set
- Caleb and Jonah: Riemann zeta function
- Will and Alex B: Polya vector fields
- Bilal and Kabir: Dirichlet problem
- Colin and Josh: hyperbolic geometry
- Derik and Jason E: branched covers and Riemann surfaces

- Week 14: Final projects
- Cassie and Jason H: fluid dynamics
- Hammad and John Michael: Fourier series
- Jasper and Joe: stationary phase method
- Logan and Alex F: elliptic functions
- Arturo and Jimmy: Schwarz-Christoffel formula
- Yizhong and Jiayuan: Gauss-Lucas theorem
- Noah and Ed: minimal surfaces
- Jeremy and Liang: elliptic functions