Math 205A: Complex Analysis (Winter, 2014)

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Announcement

Course grades are posted to SmartSite.

Solution to the final are here

Syllabus

A tentative outline of the topic we'll cover this quarter consists of the following Chapters of the text:

• Chapter 1. Preliminaries
• Chapter 2. Cauchy's theorem
• Chapter 3. Meromorphic functions and the logarithm
• Chapter 6. The Gamma and zeta functions
• Chapter 7. The prime number theorem
• Chapter 9. Elliptic functions
• Chapter 10. Theta functions

The Department syllabus is here

Final

The final exam is in-class at the registrar-scheduled time:

Fri, Mar 21, 1:00–3:00 p.m., Bainer 1060 (Exam Code R)

Here's an outline of what we covered with the corresponding sections in the text. (The usual triple-negative legal disclaimer applies: while this list is intended to be an accurate and representative reflection of what we've covered in the class, there is no guarantee that topics not listed here won't appear on the final.)

• Chapter 1. Complex Analysis (Everything)
  • Holomorphic functions
  • Cauchy-Riemann equations and d-dbar operators
  • Local behavior of holomorphic maps
  • Power series and their convergence properties
  • Holomorphic functions defined by power series
  • Definition and basic properties of contour integrals

• Chapter 2. Cauchy's theorem (We didn't cover Sec. 2.5.4--2.5.5 on the Schwarz reflection principle or Runge's theorem)
  • Bisection proof of Cauchy-Goursat theorem for triangles (or rectangles)
  • Primitives of holomorphic functions
  • Cauchy theorem for discs
  • Cauchy integral formula
  • Holomorphic functions = functions with local power series expansions
  • Liouville's theorem and applications
  • Morera's theorem and convergence/differentiation of sequences of holomorphic functions

• Chapter 3. Meromorphic functions and the logarithm (We didn't cover Sec. 3.7 on harmonic functions)
  • Isolated singularities and classification into removable, pole, or essential
  • Laurent expansions
  • Riemann surfaces and the Rieman sphere
  • Interpretation of meromorphic functions as holomorphic functions into the Riemann sphere
  • Homotopies of contours, covering spaces, winding numbers, the fundamental group, and simply connected domains
  • The homotopy form of Cauchy's theorem
  • The residue theorem and its use in calculating integrals
  • The argument principle
  • The complex logarithm

• Chapter 5. Infinite products and the sine function (Sec. 5.3 only)

• Chapter 9. Elliptic functions (Everything except Sec. 9.2.2)
• Lattices in the complex plane
• Definition and general properties of elliptic functions
• Construction of the Weierstrass P-function
• Differential equation for the P-function
• The P-function as a branched double covering of the Riemann sphere by a torus
• The parametrization of a cubic curve by the P-function
• Eisenstein series and their connection with P-functions
• Modular properties of Eisenstein series

• Chapter 10. Theta functions (Sec. 10.1 only)
• Definition and basic properties
• Product formula for the theta-function
• Connection between theta and elliptic functions

Solution to the final are here

Homework

Set 1 (Fri, Jan 17)
Ch. 1, Exercises, p. 24: 1, 4, 5, 7, 9, 10, 12, 16, 19, 23

Set 2 (Fri, Jan 24)
Ch. 1, Exercises, p. 24: 14, 15, 17, 24, 25, 26
Ch. 2, Problems, p. 67: 1

Set 3 (Fri, Jan 31)
Ch. 2, Exercises, p. 64: 1, 2, 7, 8, 9, 11, 12, 13

Set 4 (Wed, Feb 12)
Ch. 3, Exercises, p. 103: 9, 12, 16, 19, 20, 22
Ch. 3, Problems, p. 108: 1, 5

Set 5 (Wed, Feb 19)
Problems are here

Set 6 (Wed, Feb 26)
Ch. 9, Exercises p. 278: 1, 2

Set 7 (Wed, Mar 12)
Ch. 9, Exercises p. 278: 4, 6, 8
Ch. 9, Problems p. 281: 3
The additional problem here.

Set 8 (Mon, Mar 17)
Ch. 5, Exercises p. 153: 6
Ch. 10, Exercises p. 309: 1, 7