(notes courtesy of the instructor: T. Kowalski)

**Class notes:**

- Synopsis of complex numbers
- Convergence in the plane
- Topology; complex functions; complex infinity
- Complex differentiation; holomorphy
- Derivatives and integrals over curves; complex line integrals
- Complex elementary functions; Cauchy's Theorem and its consequences
- The Integral Theorems of complex analysis
- Uniqueness consequences; root functions; harmonic functions
- Power functions; circles in the plane
- Complex series; convergence tests
- Series of functions; power series and analyticity
- Uniqueness theorems; Laurent series
- Laurent expansion; singularities, poles etc.
- Residues; the Argument Principle

**Exams with full solutions:**

- Midterm, Part 1
- Midterm, Part 2; a different approach to problem 5 can be found here
- Final exam

**Important proofs:**

- Equivalence of differentiability and CR equations
- Goursat's Theorem
- Existence of antiderivatives
- Cauchy Integral Formula
- Cauchy Estimates and Liouville's Theorem
- Convergence of the Taylor Series for holomorphic functions

*Last updated: 19 August, 2007*