ABSTRACT
Harmonic functions and conformal mapping are two of the richest and most interesting topics in complex variable theory. Harmonic functions, which are always the real and imaginary parts of analytic functions, can be used to describe two-dimensional configurations of fluid flow, heat transfer, and electrical fields. Historically, conformal mapping achieved importance as a means of solving two-dimensional problems in these branches of engineering and physics. Conformal mapping provides solutions to certain canonical problems that can be used to verify the correctness of computer-generated solutions. This is analogous to the practice of learning to perform integrations in elementary calculus courses even though, for example, the MATLAB® Symbolic Mathematics Toolbox will perform them. Conformal mapping provides another means of visualizing the properties of a function of a complex variable. In this case, this chapter analyses in just two dimensions and uses two separate planes. One of the most commonly used conformal mappings in engineering is the bilinear transformation.
