Applications of Conformal Maps

Here is the payoff of the whole branch. Two facts you have already met combine into a genuinely powerful method.

First, the real and imaginary parts of a holomorphic function are harmonic — they solve Laplace's equation u_{xx} + u_{yy} = 0, the master equation of steady heat, electrostatics and ideal fluid flow. Second, a conformal map is holomorphic. Compose them and something wonderful happens:

The strategy, in three moves

Suppose you want the steady-state temperature (or electric potential) U on a complicated region D with prescribed values on the boundary.

Step 1 — transport the problem. Find a conformal map w = \phi(z) taking D onto a standard domain — say the upper half-plane or the unit disk — and carry the boundary data along with it.

U(z) = u\big(\phi(z)\big).

Step 2 — solve on the easy domain. On the half-plane or disk, Laplace's equation has clean, known solutions. For the upper half-plane with boundary values on the real axis, the Poisson integral writes u down directly; for many textbook cases u is just a linear function of the angle.

Step 3 — pull the solution back. Because harmonicity survives the conformal change of variables, U(z) = u(\phi(z)) automatically solves Laplace's equation on the original region D, with the right boundary values. The hard problem is solved without ever attacking it directly — the conformal map did the bending, and the angles (hence the orthogonal grid of equipotentials and flow lines) came through unscathed.

It is tempting to imagine conformal mapping as a universal problem-solver. It is not — and the reason is precise. The whole method rests on one fact: a harmonic function stays harmonic under a conformal change of coordinates, so a solution of Laplace's equation u_{xx} + u_{yy} = 0 transported to a simpler domain is still a solution there. That is exactly why the awkward-shape-to-disk trick works for the 2D potentials of physics: potential (irrotational, incompressible) fluid flow, steady-state heat conduction, and electrostatics.

The limitation is that this is a genuinely two-dimensional tool. Conformal maps in the rich, flexible sense used here — the holomorphic ones, with a whole Riemann mapping theorem guaranteeing that almost any planar region can be mapped to a disk — are a feature of the complex plane, and they have no counterpart in 3D. So it does not extend to a 3D heat or flow problem, and it does not rescue PDEs that are not Laplace's equation: the wave equation, the heat equation with time, or anything nonlinear (like real viscous, compressible flow) is not preserved by the map. Conformal mapping is a beautiful, powerful key — but it fits exactly one lock: 2D problems governed by Laplace's equation.

The Joukowski map and the aerofoil

The most celebrated application is in aerodynamics, through the Joukowski map

J(z) = z + \frac{1}{z}.

Its derivative J'(z) = 1 - 1/z^2 vanishes only at z = \pm 1, so J is conformal everywhere else. The magic is what it does to circles.

Step 1 — a circle centred at the origin. On the unit circle z = e^{i\theta},

J(e^{i\theta}) = e^{i\theta} + e^{-i\theta} = 2\cos\theta,

a real number sweeping the flat segment [-2, 2] — a degenerate "wing" with no thickness.

Step 2 — nudge the circle off-centre. Take instead a slightly larger circle whose centre is shifted up and to the left, but arranged to pass through the critical point z = 1. Because J'(1) = 0 is exactly where conformality fails, that one point of the circle gets pinched into a sharp cusp — the trailing edge — while the rest of the circle is mapped smoothly. The image is a curved, cambered aerofoil: rounded leading edge, sharp trailing edge.

Step 3 — solve the flow on the circle, read it on the wing. Ideal irrotational flow past a circular cylinder is a classical, solvable problem. Pushing that flow through J turns it into the flow past the aerofoil — and the lift follows from how the map redistributes the streamlines. A whole wing's aerodynamics, computed by a single complex map.

In the years around 1910 Nikolai Joukowski (and independently Martin Kutta) realised that the lift on a wing could be computed not by wrestling with the wing's true shape, but by conformally mapping the flow around a simple cylinder. The circle-to-aerofoil map J(z) = z + 1/z turned an intractable boundary into a circle, where the flow — and the circulation that creates lift — is elementary. The Kutta–Joukowski theorem, L = \rho\, V\, \Gamma, relating lift to circulation, came out of exactly this picture. Conformal mapping was, quite literally, part of the mathematics that got aircraft into the air, and Joukowski aerofoils were used on real early wings.

Bend a circle into a wing

The blue curve is a circle whose centre you move with the sliders; its radius is set so it always passes through the critical point z = 1. The orange curve is its image under J(z) = z + 1/z. Centre it at the origin and the image is the flat segment [-2, 2]; shift the centre up and to the left and the image fattens and cambers into a recognisable aerofoil, with the cusp at the trailing edge coming from z = 1.