ABSTRACT
Conformal mappings are defined, and it is shown that analytic functions are conformal mappings wherever their first derivative is nonzero. The Riemann mapping theorem is an important result. The Helmert or similarity transformation in the plane is introduced in complex notation. It is shown to be produced locally by linearization, i.e., a trucated Taylor expansion, of an analytic mapping around chosen approximate values. Generalization to three dimensions is presented, with a short remark on quaternions. Then, Möbius transformations are discussed. The generalized Liouville theorem shows that for three or more dimensions, not only every Möbius transformation is conformal, but also all conformal transformations must be Möbius transformations. Many more useful properties of Möbius transformations are presented, like the transitive property and that it maps generalized circles into generalized circles, a result that generalizes to spheres. it is shown that the stereographic projection is a restriction of a Möbius transformation. Finally, ongoing research on numerical conformal mappings is reported.
