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:
-
If u is harmonic on a domain
D' and w = \phi(z) is a conformal map
from D onto D', then the pulled-back
function U(z) = u\big(\phi(z)\big) is harmonic on
D. Harmonicity is conformally invariant.
-
So to solve Laplace's equation on an awkward domain, map it conformally to an
easy one (a disk or a half-plane), solve there, and map the answer back.
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.
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.