Georgia Institute of Technology College of Sciences

This course is designed to introduce the student to the basic results in complex variable theory, in particular Cauchy's theorem, and to develop the student's facility in the following three areas:

• Computing the Laurent or Taylor series expansions associated to a function which is analytic in part of the complex plane, and the determination of the region of convergence of such series
• Computing definite integrals by means of the residue calculus
• Solving boundary value problems associated to the Laplace operator by means of the conformal transformations associated to analytic functions

Topics to be covered:

• Geometry of the complex plane, triangle inequalities, geometric proof of the fundamental theorem of algebra
• Analytic functions: Continuity and differentiability, the Cauchy-Riemann equations. Complex functions as maps of the complex plane into itself
• Elementary analytic functions, including the logarithm, and its principle branch, log (z)
• Line integrals, the Cauchy integral formula and the Cauchy-Goursat theorem. (Proof of the Cauchy formula to be based on Green's theorem.) Morera's theorem, etc.
• Series: Taylor and Laurent expansions
• Residue calculus for definite integrals
• Harmonic functions and analytic functions: conjugate harmonics, conformal invariance of Laplace's equation in the plane, the Cauchy-Riemann equations and conformal maps
• Poisson kernel derived from Cauchy's formula; solution of boundary value problems for Laplace's equation by conformal mapping, selected applications