Math 311: Complex Analysis (Spring 2024)

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Instructor
Robert Chang

Office hours
Wednesday 2:10–3:00 p.m. and Friday 3:00–4:00 p.m. at Library 390

Grader
Valerie Wu (yuchanwu@) is our grader.

Exams
One in-class midterm
One in-person final exam on Tuesday, May 7, 1–3 p.m.

Homework
Expect weekly assignments. Work must be typeset with $\rm\LaTeX$ and submitted on GradeScope (entry code: 7DJYZW).

Collaboration, being a nontrivial part of learning and of scholarship in general, is highly encouraged. But, in accordance with the honor principle and basic human decency, you must submit your own write-up with an acknowledgment of collaborators.

Textbooks
[SS] Stein–Shakarchi, Complex Analysis

Supplements
Complex Analysis by Ahlfors
Complex Analysis by Gamelin
Math 311 class notes by Jerry Shurman

Game Plan
We hope to cover chapters 1, 2, 3, 6, as well as parts of chapters 7 and 8, of [SS].
Here is a syllabus with all the legalese.
Week
Date
Topics covered and suggested readings
1
1/22
Complex numbers: arithmetic, geometry, polar form
Review on your own: basic topology (onterior points, open sets, compactness, etc.) and analysis (convergence, continuity, etc.)
[SS, pp. 1–7]
 
1/24
Complex differentiability, Cauchy–Riemann equations, holomorphic functions
[SS, pp. 8–13]
 
1/26
Power series
[SS, pp. 14–18]
Homework 1 (pdf, tex, solutions) due Friday, February 2
2
1/29
Line integrals
[SS, pp. 19–24]
 
1/31
Goursat's theorem, Cauchy's theorem for a disc
[SS, pp. 32–41]
 
2/2
Evluation of some integrals using countour integration
[SS, pp. 41–45]
Homework 2 (pdf, tex, solutions) due Friday, February 9
3
2/5
The Cauchy integral formulas and Cauchy inequalities
[SS, pp. 45–50]
 
2/7
Corollaries: Liouville, Morera, Schwarz reflection
[SS, pp. 50–53, 57–60]
 
2/9
Corollaries: Liouville, Morera, Schwarz reflection
[SS, pp. 50–53, 57–60]
Homework 3 (pdf, tex, solutions) due Friday, February 16
4
2/12
Removable singularities, zeros
[SS, pp. 71–74, 83–85]
 
2/14
Poles and residues
[SS, pp. 74–77]
 
2/16
Poles and residues
[SS, pp. 74–77]
Homework 4 (pdf, tex, solutions) due Friday, February 23
5
2/19
The residue theorem, examples
[SS, pp. 77–87]
 
2/21
Hyperbolic functions and another example of residue calculus, essential singularities
[SS, pp. 86–89]
 
2/23
Riemann sphere, singularity at infinity
[SS, pp. 86–89]
Homework 5 (pdf, tex, solutions) due Friday, March 1 March 8
6
2/26
Argument principle, Rouché's theorem
[SS, pp. 89–91]
 
2/28
Open mapping theorem, maximum modulus principle
[SS, pp. 92–93]
 
3/1
Deformation of countour
[SS, pp. 93–97]
Homework 5 (pdf, tex, solutions) due Friday, March 8
7
3/4
Complex logarithm
[SS, pp. 97–101]
 
3/6
In-class midterm Uniform convergence on compact subsets
[SS, pp. 53–57]
 
3/8
Abstract harmonic analysis
Spring break
Next homework due Friday, March 29
9
3/18
Fourier transform
[SS, pp. 111–118]
 
3/20
Fourier transform
[SS, pp. 111–118]
 
3/22
Poisson summation
[SS, pp. 118–120]
Homework 6 (pdf, tex, solutions) due Friday, March 29
10
3/25
Paley–Weiner theorem
[SS, pp. 121–126]
 
3/27
Infinite products
[SS, pp. 140–144]
 
3/29
Gamma function: meromorphic continuation and functional equation
[SS, pp. 159–168]
Homework 7 (pdf, tex, solutions) due Friday, April 5
11
4/1
Gamma function: Wielandt's theorem, Euler's definition of $\Gamma$
[SS, pp. 159–168]
 
4/3
Gamma function: Weierstrass's definition of $\Gamma$, logarithmic derivative
[SS, pp. 159–168]
 
4/5
Laplace transform and Stirling's formula
[SS, pp. 323–328]
Homework 8 (pdf, tex, solutions) due Friday, April 12
Next Homework due Friday, April 19
12
4/8
Crash course on Dirichlet series, Zeta function: definition and Euler's product formula
[SS, pp. 168–174]
 
4/10
Zeta function
[SS, pp. 168–174]
 
4/12
Zeta function: meromorphic continuation, functional equation, Prime number theorem
Show comments
Homework 8 (pdf, tex, solutions) due Friday, April 19
13
4/15
No class
 
4/17
Prime number theorem
 
4/19
Prime number theorem
Homework 9 (pdf, tex, solutions) due Friday, April 26
14
4/22
Prime number theorem
 
4/24
Prime number theorem
 
4/26
No class
In-person final exam on Tuesday, May 7, 1–3 p.m. at Phys 240A
Focus on the fundamentals: Complex differentiation, Cauchy–Riemann equation, explicit computations of line integrals by parametrization, Cauchy's theorem, Cauchy's integral formulas and inequalities, Liouville's theorem, residue formula, etc.

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