Complex Variables
Math 421 Fall 2002
Syllabus

• 1) Complex Numbers: algebraic and geometric properties, polar form, powers and roots.

• 2) Analytic functions: Differentiability and Cauchy-Riemann equations, Harmonic functions, examples.

• 3) Elementary functions of a complex variable: exponential and trigonometric functions, logarithms.

• 4) Path integrals: contour integration and Cauchy's integral formula; Liouville's theorem, Maximum modulus theorem, the Fundamental Theorem of Algebra.

• 5) Series: Taylor and Laurant expansions, convergence, term-by-term operations with infinite series.

• 6) Isolated singularities and residues. Essential singularities and poles. Evaluation of Improper integrals via residues.

• 7) (If time permits) Mappings by elementary functions and linear fractional transformations; conformal mappings.

maintained by Eyal Markman