MATH 335 Homepage (Fall 2016)

Lecturer: Alexandru Ionescu,


Problem Set 1. Due in class on Thursday , September 22.
Problem Set 2. Due in class on Thursday , September 29.
Problem Set 3. Due in class on Thursday , October 6.
Problem Set 4. Due in class on Thursday , October 13.
Problem Set 5. Due in class on Tuesday , October 25.
Problem Set 6. Due in class on Tuesday , November 15.
Problem Set 7. Due in class on Tuesday , November 22.
Problem Set 8. Due in class on Thursday , December 1.
Problem Set 9. Due in class on Thursday , December 8.
Problem Set 10. Due in class on Friday, December 16.


Sep. 15: Introduction to Complex Analysis, holomorphic functions.
Sep. 16: The Cauchy-Riemann equations, power series.
Sep. 20: Integration along curves, Goursat's theorem.
Sep. 22: The Cauchy theorem, Cauchy's integral formulas, analytic functions.
Sep. 23: Applications: Liouville's theorem, the fundamental theorem of algebra, Morera's theorem, analytic continuation.
Sep. 27: Sequences of holomorphic functions, Schwarz reflection principle.
Sep. 29: Removable singularities, poles, the residue formula.
Oct. 04: The argument principle, Rouche's theorem, Cauchy's theorem in simply connected domains.
Oct. 06: The complex logarithm, Fourier series, the Riemann sphere.
Oct. 07: The Fourier transform: the Fourier inversion formula and Plancherel theorem.
Oct. 11: The Fourier transform on holomorphic functions.
Oct. 13: The Poisson summation formula.
Oct. 14: The Paley-Wiener theorem.
Oct. 18: Entire functions: Jensen's formula.
Oct. 20: Functions of finite order, infinite products.
Oct. 21: Weierstrass infinite products, examples.
Oct. 25: Hadamard's factorization theorem.
Oct. 27: Midterm exam.
Oct. 28: The Gamma function: definition and basic properties.
Nov. 08: The Gamma function: meromorphic extension and further results.
Nov. 10: The zeta function: definition and the functional equation.
Nov. 11: The zeta function: meromorphic extension and bounds.
Nov. 15: The Prime Number Theorem: outline of the proof.
Nov. 17: The zeta function in the critical strip.
Nov. 18: Proof of the Prime Number Theorem.
Nov. 22: Conformal mappings, examples.
Nov. 29: The Schwarz lemma, automorphisms of the disk, the Riemann mapping theorem.
Dec. 01: Proof of the Riemann mapping theorem.
Dec. 02: Conformal mappings onto polygons: extension to the boundary.
Dec. 06: Conformal mappings onto rectangles.
Dec. 08: Elliptic functions: definitions and main properties.
Dec. 09: The Weierstrass p function.
Dec. 13: Theta functions: definitions and basic properties.
Dec. 15: The two-squares theorem: main ideas.
Dec. 16: Proof of the two-squares theorem.