Some notes from an undergraduate complex analysis course
(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

Back to R. Virk's homepage

---

Last updated: 19 August, 2007