Math 440/508 - Complex Analysis - Fall 2016

Instructor: Malabika Pramanik
Office: 214 Mathematics Building
E-mail: malabika at math dot ubc dot ca
Lectures: Mon,Wed,Fri 11:00 AM to 12:00 noon in Room 105 of Mathematics Building.
Office hours: Wed 10-11AM, Fri 12-1PM or by appointment.

Course information

• Course information handout containing syllabus and grading policies.

Homework

• Homework 1, due on Monday September 19
  • Chapter 1 exercises: 4, 5, 7, 21, 22.
• Homework 2, due on Monday October 3
  • Chapter 2 exercises: 1, 2, 5, 7, 8, 9, 10, 1, 12, 13.
• Homework 3, due on Friday October 21
  • Chapter 2 exercises: 14, 15
  • Chapter 2 problems: 1, 3, 5.
• Midterm, due on Wednesday November 2
  • Chapter 3 exercises: 14, 15, 16, 21
  • Chapter 3 problems: 3.
• Homework 4, due on Wednesday November 23
  • Chapter 8 exercises: 5, 13, 14.
  • Chapter 8 problems: 3, 4.
• Take-home final, due by noon on Monday December 5
  • Chapter 8 exercises: 19.
  • Chapter 8 problems: 5, 7, 8,
  • Please leave your final exam in the instructor's mailbox or slide under the office door.

Week-by-week course outline

Here is a rough guideline of the course structure, arranged by week. The textbook pages are mentioned as a reference and as a reading guide. The treatment of these topics in lecture may vary somewhat from that of the text.

• Week 1: (pages 1-18)
  • Definition of holomorphy
  • Cauchy-Riemann equations
  • Power series
• Week 2: (pages 21-36)
  • Integration along curves
  • Goursat's theorem
• Week 3: (pages 37-50)
  • Cauchy's theorem on the disc
  • Toy contours
  • Cauchy integral formulae
• Week 4: (pages 50-53, 41-45)
  • Power series representation of holomorphic functions
  • Liouville's theorem
  • Fundamental theorem of algebra
  • Principle of analytic continuation
  • Morera's theorem
  • Evaluation of some integrals
  • Morera's theorem
• Week 5: (pages 71-87)
  • Morera's theorem
  • Extended complex plane
  • Isolated singularities
  • Removable, pole and essential singularities
• Week 6: (pages 89-93)
  • Residue theorem
  • Argument principle
  • Homotopies and simply connected domains
• Week 7: (pages 93-101)
• Cauchy's theorem on homotopies
• The complex logarithm
• Week 8: (pages 205-212, 218)
• Conformal mappings
• Schwarz lemma
• Automorphisms of the disc
• Week 9:
• Mobius transformations
• Cross-ratio
• Preservation of generalized circles