Complex Variables (Applied MS)

Preparatory Course: Math 4283

1.

Complex numbers and functions: Arithmetic of complex numbers, de Moivre's theorem, topology of the complex plane, path-connectedness, domains, limits, continuity, Cauchy-Riemann equations, harmonic and analytic functions

2.

Elementary functions and mappings: Exponential, trigonometric, and logarithmic functions, roots, branches, linear fractional transformations, conformal maps

3.

Contour integration: Line integrals, Cauchy-Goursat theorem, Cauchy integral formula, Morera's theorem, maximum modulus principle

4.

Series: Sequences and series of complex numbers and functions, uniform convergence, Taylor's series, Laurent's series, isolated singularities, residues

5.

Applications: Evaluation of integrals of real- valued functions, Laplace's equation, fluid flows, temperature distributions, electrostatic potential

REFERENCE: Complex Variables and Applications, (6th edition) by R. V. Churchill and J. W Brown, McGraw-Hill Book Company, 1995.