the Joukowski map
/ zhoo-KOW-skee /
Here is a conformal map with an unusually concrete claim to fame: it turns circles into airfoil shapes, the cross-sections of an aeroplane wing. Long before computers could simulate airflow, engineers used this single elegant formula to design wings whose lift they could actually compute by hand. The Joukowski map is the bridge between the easy geometry of a circle and the hard geometry of a wing.
Its formula is w = (1/2)(z + 1/z), often written without the 1/2 as w = z + 1/z. To see its character, set z on the unit circle, z = e^(i theta); then w = (1/2)(e^(i theta) + e^(-i theta)) = cos theta, which sweeps the real segment from -1 to 1 as theta runs around. So the Joukowski map flattens the unit circle onto a line segment (a 'degenerate airfoil'). Its derivative w' = (1/2)(1 - 1/z^2) vanishes at z = 1 and z = -1, so those are critical points; the sharp trailing edge of a Joukowski airfoil comes precisely from the angle-doubling at the critical point z = 1. Take a circle that passes through z = 1 but is slightly offset and enlarged, and its image is a smooth wing-shape with a cusp at the trailing edge. The map is two-to-one in general (z and 1/z give the same w), so you must restrict to the outside (or inside) of the unit circle to make it a genuine conformal bijection.
Its importance is twofold. Mathematically, it is the model example of how a critical point manufactures a corner, and (restricted to the exterior of the unit disk) it conformally maps that exterior onto the plane slit along [-1, 1]. Physically, it lets you transplant the simple, known flow of an ideal fluid around a circular cylinder into the flow around a wing — and from that flow, via the Kutta-Joukowski theorem, you read off the lift. A caution: the airfoil story is an idealization (inviscid, incompressible flow); real wings involve viscosity and the boundary layer that this clean complex-analysis picture leaves out.
On the unit circle z = e^(i theta), the map w = (1/2)(z + 1/z) gives w = cos theta — the entire circle collapses onto the real segment [-1, 1]. A circle passing through z = 1 but centered slightly off the origin instead maps to a curved airfoil with a sharp cusp where it crossed z = 1.
The Joukowski map flattens the unit circle to a segment; offset circles become airfoils.
Because z and 1/z share an image, the Joukowski map is two-to-one on the whole plane; it is conformal and injective only after you restrict to one side of the unit circle and stay away from the critical points z = +-1.