conformal mapping

Imagine printing a grid of tiny squares on a rubber sheet, then stretching and bending the sheet into a new shape. A conformal map is a deformation that, no matter how it warps the overall picture, preserves angles in the small: each tiny square is rotated and resized but never sheared — it stays a square. This angle-preserving property is exactly what an analytic function with nonzero derivative does, and it is the bridge that turns a hard-shaped physics problem into an easy-shaped one.

Precisely, a map w = f(z) is conformal at a point where f is analytic and f'(z) is not zero. There f acts locally like multiplication by the complex number f'(z): it rotates every direction by the argument of f'(z) and scales every length by the magnitude of f'(z), the same for all directions, which is why angles between curves are preserved. Where f' = 0 the map fails to be conformal and angles can be doubled or worse. The central application rests on a fact you already know: harmonic functions stay harmonic under a conformal change of variable, so a solution of Laplace's equation on a complicated region can be pulled back from a solution on a simple region.

This is the classical engine of two-dimensional potential theory. To find the electrostatic potential, ideal fluid flow, or steady temperature in an awkwardly shaped domain, you conformally map the awkward region onto a simple one — a half-plane or a disk — where the answer is known, solve there, and map back. Mobius transformations map disks and half-planes to one another; the Joukowski map turns a circle into an airfoil and underlies classical wing theory. The Riemann mapping theorem promises that essentially any simply connected region can be mapped to the disk, though finding the explicit map can be hard.