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.