transplanting a problem by conformal mapping

Suppose you must find the steady temperature inside an oddly shaped plate, or the electric field around a strangely curved electrode. The geometry is so awkward that writing the answer directly looks hopeless. The trick is to move the whole problem to a region where the answer is easy — a disk, a half-plane, a straight strip — solve it there, and then carry the solution back. A conformal map is the vehicle that carries it, and it does so without corrupting the physics, because the physics here is governed by angles.

The reason this works is a small miracle: if h is a harmonic function (a solution of Laplace's equation h_xx + h_yy = 0) on a region, and f is a conformal map from a second region onto the first, then the composition h(f(z)) is again harmonic on the second region. Harmonicity survives the conformal change of variables. So the recipe is: (1) find a conformal map f from your easy region E onto your hard region D; (2) solve the (easy) problem on E — often you can just write the harmonic function down; (3) compose with f to pull the solution back to D; (4) translate the boundary conditions along the way, since f matches up the boundaries. The classic case is the Dirichlet problem: prescribed boundary temperatures whose interior values you want.

This is the single most powerful application of conformal mapping, and the reason the Riemann mapping theorem is treasured — it promises that almost any simply connected region can be mapped conformally onto the unit disk, where the Dirichlet problem is solved once and for all by the Poisson integral formula. Two honest cautions, though. First, not every quantity transplants: lengths, areas, and the Laplacian's actual size all change, so you must track the conformal factor |f'| for anything that is not angle-based. Second, finding the map f explicitly can be the hard part; the theorem guarantees it exists but does not hand it to you.