the Schwarz-Christoffel formula
/ shvarts KRIS-toh-fel /
Mobius maps reshape disks and half-planes into other disks and half-planes, and exp and log handle wedges — but what if the region you care about is a polygon, a shape with straight sides and sharp corners, like a rectangle, an L-shaped room, or a channel with a step in it? The Schwarz-Christoffel formula is the explicit recipe that builds a conformal map from the upper half-plane (or the disk) onto the interior of any polygon. It is the tool of choice for polygonal regions, which arise constantly in electrostatics and fluid flow through ducts.
The idea behind the formula is to engineer the corners. A conformal map turns angles by arg f', and a critical-point-style factor (z - x_k)^(power) bends the boundary by a controlled amount as you pass the point x_k. So you place points x_1 < x_2 < ... < x_n on the real axis (these are the pre-images of the polygon's vertices) and multiply one bending factor per vertex. The map is given as an integral: f(z) = A + C * integral from a base point to z of the product over k of (w - x_k)^(alpha_k - 1) dw, where pi * alpha_k is the interior angle of the polygon at the k-th vertex (so alpha_k - 1 is negative for a convex corner). Integrating accumulates exactly the right turning to trace out the polygon's boundary, straight edge by straight edge with a kink of the prescribed angle at each vertex. The constants A and C just position, rotate, and scale the resulting polygon.
This is a genuine constructive realization of the Riemann mapping theorem for polygons — it actually writes the map down, where the general theorem only promises existence. Two honest difficulties keep it from being a free lunch. First, the integral usually cannot be evaluated in elementary terms (for a rectangle it produces elliptic integrals), so the map is often computed numerically. Second, and more subtly, the points x_k on the real axis are not free to choose — they must be solved for so that the resulting side lengths match the target polygon, and this 'parameter problem' is a nonlinear system that is awkward to solve by hand. Despite this, Schwarz-Christoffel remains the standard route to conformal maps of polygonal domains.
To map the upper half-plane to an infinite strip 0 < Im w < pi (a degenerate polygon with two vertices at infinity), the formula collapses to f(z) = log z (up to constants): exactly the log map that unrolls a sector. The single bending factor encodes the strip's two right-angle 'corners' sent off to infinity.
Each vertex contributes one factor (z - x_k)^(alpha_k - 1); integrating traces the polygon's boundary.
The hard part is the 'parameter problem': the pre-vertex positions x_k cannot be chosen freely but must be found so the side lengths come out right. You may fix three of them (using the Mobius freedom), but the rest must be solved numerically.