Instructor
Office Hours
Grader
Textbooks
Supplements
Resources
Exams
Homework
Robert Chang (rchang@reed.edu)
Wednesday 2:10–3:00 p.m. and Friday 3:00–4:00 p.m. at Library 390
Valerie Wu (yuchanwu@)
[SS] Stein–Shakarchi, Complex Analysis
[Ga] Gamelin, Complex Analysis
[GK] Greene and Krantz, Function Theory of One Complex Variable
Ahlfors, Complex Analysis
Math 311 class notes by Jerry Shurman
Math Help Center/Drop-in Tutoring: SuMTTh 7:00–9:00 p.m. at Library 204
Individual Tutoring: Reed offers one hour per week of free, one-to-one tutoring
One in-class midterm
One in-person final exam on Tuesday, May 7, 1–3 p.m.
Expect weekly assignments. Work must be typeset with 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.
Week
1
Date
1/22
1/24
1/26
Topics and Reading
Complex numbers: arithmetic, geometry, polar form
Review on your own: basic topology and analysis
Read [SS, pp. 1-7]
Complex differentiability, Cauchy–Riemann equations, holomorphic functions
Read [SS, pp. 8-13]
Power series
Read [SS, pp. 14-18]
Homework 1 (pdf, tex, solutions) due Friday, February 2
2
1/29
1/31
2/2
Line integrals
Read [SS, pp. 19-24]
Goursat's theorem, Cauchy's theorem for a disc
Read [SS, pp. 32-41]
Evaluation of some integrals using contour integration
Read [SS, pp. 41-45]
Homework 2 (pdf, tex, solutions) due Friday, February 9
3
2/5
2/7
2/9
The Cauchy integral formulas and Cauchy inequalities
Read [SS, pp. 45-50]
Liouville's theorem, zeros and the uniqueness principle, Morera's theorem
Read [SS, pp. 50-53]
Sequences of holomorphic functions, holomorphic functions defined in terms of integrals
Read [SS, pp. 53-57]
Homework 3 (pdf, tex, solutions) due Friday, February 16
4
2/12
2/14
2/16
Classification of singularities, Riemann's theorem on removable singularities, Casoratti–Weierstrass
Read [GK, 105-109]
The presentation in [SS] is too bare bones; we will follow [GK, § 4] instead.
Another excellent source with more details is [Ga, §§ 5-7].
Laurent expansions: existence, uniqueness, examples
Read [GK, pp. 109-122]
Laurent expansions continued
Read [GK, pp. 109-122]
Homework 4 (pdf, tex, solutions) due Friday, February 23
5
2/19
2/21
2/23
The residue theorem, applications of residue calculus
Read [SS, pp. 76-83], [GK, pp. 128-137]
I present only the "vanilla version" (i.e., not the one from [GK, § 4.5]) of the residue theorem.
See [Ga, §§ 7.1-7.6] for more worked examples.
More examples of residue calculus, Jordan's lemma
Read [SS, pp. 76-83], [Ga, pp. 216-218], [GK, pp. 128-137]
Meromorphic functions and singularities at infinity, the Riemann sphere
Read [SS, pp. 86-88], [GK, pp. 137-145]
Homework 5 (pdf, tex, solutions) due Friday, March 1 March 8
6
2/26
2/28
3/1
Argument principle, Rouché's theorem
Read [SS, pp. 89-91]
Open mapping theorem, maximum modulus principle
Read [SS, pp. 89-91]
Deformation of contour
Read [SS, pp. 93-97]
Homework 5 (pdf, tex, solutions) due Friday, March 8
7
3/4
3/6
3/8
The complex logarithm, inverse of trigonometric functions
Read [SS, 97-101]
In-class midterm
Crash course on abstract harmonic analysis on LCA groups
Spring break
Next homework due Friday, March 29
9
3/18
3/20
3/22
The Fourier transform
Read [SS, pp. 111-118]
The Fourier transform
Read [SS, pp. 111-118]
The Poisson summation formula, transformation law for the theta function
Read [SS, pp. 118-120]
Homework 6 (pdf, tex, solutions) due Friday, March 29
10
3/25
3/27
3/29
The Paley–Wiener theorem
Read [SS, pp. 121-126]
The Paley–Wiener theorem continued, infinite products
Read [SS, pp. 121-126, 140-144]
The Gamma function: meromorphic continuation and functional equation
Read [SS, pp. 159-168], [Ga, pp. 361-364], [GK, pp. 449-457]
Click to show additional comments on our study of special functions
As you might expect, the theory of these special functions is very classical and well-studied. I will synthesize various sources to give a detailed presentation of the Gamma function. I will then discuss two methods---one from [SS] using the theta function, and another from [Ga, GK] using contour integration---of meromorphically extending the Zeta function. Technical estimates found in [SS] used to prove the prime number theorem will be omitted, as we will employ an alternative proofs strategy that is found in [Ga, GK].
Homework 7 (pdf, tex, solutions) due Friday, April 5
11
4/1
4/3
4/5
The Gamma function: Wielandt's theorem, Euler's definition of Γ
Read [SS, pp. 159-168], [Ga, pp. 361-364], [GK, pp. 449-457]
Gamma function: Weierstrass's definition of Γ, the logarithmic derivative
Read [SS, pp. 159-168], [Ga, pp. 361-364], [GK, pp. 449-457]
Laplace transform and Stirling's formula
Read [Ga, pp. 365-369]
Next Homework due Friday, April 19
12
4/8
4/10
4/12
Crash course on Dirichlet series, Zeta function: definition and Euler's product formula
Read [Ga, pp. 370-371, 376-379], [GK, 457-458]
Zeta function: meromorphic continuation method 1
Read [Ga, pp. 372-375], [GK, 459-463]
Zeta function: meromorphic continuation method 2
Read [SS, pp. 168-172]
Homework 8 (pdf, tex, solutions) due Friday, April 19
Click to show additional comments on our proof of the prime number theorem
The proof in [SS, pp. 168–174] contains many technical estimates. I will present an alternative, less technical version from [Ga, GK]; see also Zagier's short exposition of Newman's proof.
13
4/15
4/17
4/19
Class canceled
Prime number theorem
Read [Ga, pp. 372-375], [GK, 459-463]
Prime number theorem
Read [Ga, pp. 382-388], [GK, 471-484]
No Homework 9
14
4/22
4/24
4/26
Prime number theorem
Read [Ga, pp. 372-375], [GK, 459-463]
Prime number theorem
Read [Ga, pp. 382-388], [GK, 471-484]
Class canceled
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.