Math 494, Spring 2013

Welcome to Math 494, Independent Study in Complex Analysis.

• Instructor: Po Lam Yung
  Office: Hill 224

Textbook and Plan:

• Our plan is to go through Complex Analysis by Elias M. Stein and Rami Shakarchi.
• We meet twice a week, where students take turn to present the subject matter proper. The presentation will be followed up by some in-depth discussion where every student is expected to participate. There will be an additional student-run problem solving session that meets once a week.
• Below is a tentative schedule for the presentations:

Week 1: Chapter 1

• Topics:
• Definition of holomorphic functions
• Cauchy-Riemann equations
• Functions defined by a convergent power series are holomorphic
• Contour integrals
• Homework:
• Exercise 1(a)(b)(d)(g), 4, 5, 7, 8, 10, 13, 21, 22, 25 from Chapter 1

Week 2: Chapter 2.1-2.4

• Topics:
• Goursat's theorem and local existence of primitives
• Cauchy's theorem for a disk and example
• Cauchy integral formulas
• Expanding holomorphic functions as power series
• Homework:
• Exercise 15, 17, 19, 23 from Chapter 1
• Exercise 1, 3, 4, 5, 7 from Chapter 2

Week 3: Chapter 2.5-3.1

• Topics:
• Morera's Theorem and Sequence of holomorphic functions
• Schwarz reflection principle
• Runge's Approximation Theorem
• Zeros and Poles
• Homework:
• Exercise 10, 11, 12, 13, 14, 15, and Problem 2, from Chapter 2

Week 4: Chapter 3.2-3.4

• Topics:
• Residue Theorem
• Description of Isolated Singularities
• Argument Principle and Rouche's Theorem
• Open Mapping and Maximum Modulus Principle
• Homework:
• Exercise 1, 3, 8, 9, 11, 12, 13, 14 from Chapter 3

Week 5: Chapter 3.5-3.7

• Topics:
• Homotopies and Simply connected domains
• Complex Logarithms defined via contour integrals
• Fourier series
• Harmonic functions
• Homework:
• Exercise 15, 16, 17, 19, 20, 22 and Problem 3 from Chapter 3

Week 6: Chapter 4

• Topics:
• The Fourier transform
• Functions analytic in a strip
• Poisson summation formula
• Paley-Wiener Theorems
• Homework:
• Exercise 1, 3, 4, 6, 7, 9, 12 and Problem 3 from Chapter 4

Week 7: Chapter 5

• Topics:
• Jensen's formula and functions of finite order
• Infinite Products
• Weierstrass Factorization
• Hadamard's Factorization for functions of finite order
• Homework:
• Exercise 3, 4, 5, 6, 8, 10, 11, 13 from Chapter 5

Week 8: Chapter 6

• Topics:
• Gamma Function and its Analytic Continuation
• Factorization of 1/Gamma
• Zeta Function and its Functional Equation
• Analytic Continuation of Zeta, and Growth of Zeta on the line Re s = 1
• Homework:
• Exercise 1, 4, 6, 11, 12, 15, 16 and Problem 1(a)(b) from Chapter 6

Week 9: Chapter 7

• Topics:
• Zeros of Zeta outside the strip 0 < Re s < 1
• Growth of 1/Zeta on the line Re s = 1
• Proof of Prime Number Theorem: Reduction to estimates for Psi_1
• Integral representation of Psi_1, and Estimates for Psi_1
• Homework:
• Exercise 1, 2, 3, 5, 8, 11, 12 and Problem 2 from Chapter 7

Week 10: Chapter 8.1-8.3

• Topics:
• Conformal Equivalence and Examples
• Schwarz Lemma and Automorphism of the Disc
• Automorphism of the upper half space
• Montel's Theorem
• Homework:
• Exercise 1, 4, 5, 10, 11, 12, 13 from Chapter 8

Week 11: Chapter 8.3, 9.1

• Topics:
• Riemann mapping theorem: statement and proof
• Elliptic functions; zeroes and poles
• Weierstrass ℘ function
• Properties of ℘
• Homework:
• Problem 3 and 4 from Chatper 8
• Exercise 1, 2, 3, 4 from Chapter 9

Week 12: Chapter 9.2, 10.1

• Topics:
• Eisenstein series
• Relation to divisor functions
• The Jacobi Theta function and the product formula
• Further properties of Theta
• Homework:
• Exercise 5, 6, 7, 8 from Chapter 9
• Exercise 1 from Chapter 10 (Correction: "First two derivatives" in Exercise 1 should read "First three derivatives")

Week 13: Chapter 10.2, 10.3

• Topics:
• Partition functions
• Sum of two squares theorem: Introduction
• Sum of two squares theorem: Conclusion of proof
• Sum of four squares theorem
• Homework:
• Exercise 5, 7, 9, 11 and Problem 2 from Chapter 10