MATH423

Perform basic algebraic operations using various representations of the complex number system.

Prove simple facts about the topology of the complex number system and functions of a single complex variable.

Compare and contrast functions of a real variable and functions of a complex variable.

Identify and differentiate analytic functions.

Find harmonic functions that solve classical applied problems in mathematics.

Define complex generalizations of the definite integral of a function of a real variable.

Compute contour integrals.

State the Cauchy Integral Formula and Liouville's Theorem.

Compute the Taylor and Laurent expansions of simple functions, determining the nature of the singularities, regions of convergence, and calculating residues.

Use the Cauchy Residue Theorem to evaluate integrals.

Apply ideas and methods of complex analysis to problems in science and engineering.