Math 185A: Complex Analysis

Important Dates

Final

Solutions to the final exam

Some sample final questions

Outline of topics

• Chapter 1
  • Complex numbers
  • Exponential, trigonometric, logarithmic and root functions
  • Branch cuts
  • The Riemann sphere
  • Limits and continuity
  • Analytic functions and the complex derivative
  • The Cauchy-Riemann equations
  • Harmonic functions
• Chapter 2
  • The definition and evaluation of contour integrals
  • Estimating contour integrals (Proposition 2.1.6)
  • Path independence and antiderivatives
  • Cauchy's theorem for simple closed curves
  • Homotopy version of Cauchy's theorem
  • Winding number
  • Cauchy integral formula for an analytic function and its derivatives
  • Liouville's theorem
  • Maximum modulus principle
  • Mean value property of analytic and harmonic functions
  • Poisson's formula for harmonic functions on a disc
• Chapter 3
  • Uniform and absolute convergence; Weierstrass M-test
  • Power series of an analytic function; radius of convergence
  • Differentiation of power series and Taylor's theorem
  • Power series expansions of common functions (Table 3.2.1)
  • Laurent expansion of functions analytic in an annulus
  • Isolated singularities
  • Poles, removable singularites, and essential singularities
  • The residue of a function at an isolated singularity
  Chapter 4
  • Computation of residues
  • The residue theorem for simple closed curves
  • Homotopy version of the residue theorem
  • Evaluation of definite integrals by the method of residues (principal value integrals and integrals involving branch cuts will not be on the final)
  • Evaluation of series by the method of residues (partial-fraction expansions will not be on the final)
  Chapter 5 (5.1 only)
  • Conformal mappings
  • Riemann mapping theorem
  • Simple examples

  Some Important Theorems

  You don't need to memorize the proofs, but you should be be able to state the theorems and know how to use them.
  

  • Theorem 1.5.7: Conformal Mapping Theorem
  • Theorem 1.5.8: Cauchy-Riemann Theorem
  • Theorem 1.5.10: Inverse Function Theorem
  • Theorem 2.1.7: Fundamental Theorem of Calculus for Contour Integrals
  • Theorem 2.1.9: Path Independence Theorem
  • Theorem 2.2.3: Cauchy's Theorem for a Simply Connected Region
  • Theorem 2.3.14: Homotopy Form of Cauchy's Theorem
  • Theorem 2.4.4: Cauchy Integral Formula
  • Theorem 2.4.6: Cauchy Integral Formula for Derivatives
  • Theorem 2.5.1: Maximum Modulus Principle
  • Theorem 2.5.2: Mean Value Property for Analytic Functions
  • Theorem 2.5.9: Mean Value Property for Harmonic Functions
  • Theorem 2.5.13: Poisson's Formula (you don't need to memorize it, but you should understand what it says)
  • Theorem 3.1.7: Weierstrass M Test
  • Theorem 3.2.3: Analyticity of Power Series
  • Theorem 3.2.4: Differentiation of Power Series
  • Theorem 3.2.7: Taylor's Theorem
  • Theorem 3.3.1: Laurent Expansion Theorem
  • Theorem 4.2.1: Residue Theorem
  • Theorem 5.1.4: Riemann Mapping Theorem

Midterm

Midterm solutions are here.

Here is a sample midterm from a previous MAT 185 taught by another instructor. Solutions are included, but I recommend that you try the problems yourself first before looking at the solutions. A couple of questions have topics from Section 2.5 (e.g. the maximum modulus principle) that won't be on our exam, so you can ignore those parts.

Exams

There will be one in-class midterm and a final.

• Midterm: Wednesday, February 10
• Final: Tuesday, March 16, 3:30–5:30 p.m. (Exam Code D)

Grade

Grade will based on exams and homework, weighted

• 25%: Homework
• 35%: Midterm
• 40%: Final

Mobius transformations

Video from Institute of Mathematics and its Applications

Homework solutions

Solutions for even numbered problems

Homework (Due in class)

• Set 1 (Wednesday, January 13)
  Sec 1.1, p. 9: 2, 13
  Sec 1.2, p. 22: 1, 8, 9, 19

  Set 2 (Wednesday, January 20)
  Sec 1.3, p. 38: 2, 5, 9, 10, 21, 22
  Read Sec. 1.4
  Sec 1.4, p. 57: 1, 3, 6, 8, 11, 14, 20, 21

  Set 3 (Wednesday, January 27)
  Sec 1.5, p. 77: 2, 9, 10, 27, 30
  Sec 1.6, p. 89: 3, 6, 9, 14
  Review, p. 90: 17, 18

  Set 4 (Wednesday, February 3)
  Sec 2.1, p. 109: 1(a), 5, 8, 10, 11(b), 14
  Sec 2.2, p.122: 1(c), 5, 7, 8

  Set 5 (Wednesday, February 10)
  Sec 2.4, p.161: 3, 5, 7, 15, 17, 21

  The empty set (Wednesday, February 17)

  Set 7 (Wednesday, February 24)
  Sec 2.5, p.176: 1, 3, 5, 10, 12, 18
  Sec 3.1, p. 201: 1, 8, 17
  Sec 3.2, p. 218: 1, 3, 6

  Set 8 (Wednesday, March 3)
  Sec 3.3, p. 234: 1, 5, 8, 14, 16, 19, 20

  Set 9 (Wednesday, March 10)
  Sec 4.1, p. 254: 2, 5, 9, 11
  Sec 4.2, p. 268: 3, 6, 9
  Sec 4.3 p. 301: 1, 4, 7

  Set 10 (Monday, March 15: will not be collected)
  Sec 4.4 p. 312: 1
  Sec 5.1 p. 325: 3, 5
  Sec 5.2 p. 342: 1, 5