Joukowski Mapping of Circular Domains to Aerodynamic Profiles

Conformal mapping is one of the central geometric techniques of complex analysis because it permits a complicated planar domain to be represented by a simpler one while preserving local angles. This paper develops the analytic conditions for conformality from the principal linear part of a differentiable complex mapping, relates the Cauchy--Riemann equations to local rotation and dilation, and presents explicit mappings between standard domains. Particular attention is given to the Joukowski transformation and its constructive action on circular boundaries. Actual mappings are worked out for the upper half-plane and unit disk, for a strip and half-plane, and for circles mapped by the Joukowski transformation into a line segment, an ellipse, and an airfoil-like profile. The derivations are accompanied by graphical realizations and a discussion of potential-flow aerodynamics. The results make explicit the connection between the analytic formula, its critical points, transformed geometry, and the distinction between local conformality and global one-to-one behavior.