the Schwarz-Christoffel mapping
/ shvarts KRIS-toh-fel /
The Riemann mapping theorem promises a conformal map onto the disk but hands you no formula. The Schwarz-Christoffel mapping is the glorious exception: when the target region is a POLYGON, there is an explicit integral formula for the conformal map. It is the workhorse that turns the abstract existence theorem into something you can actually compute and use, mapping the upper half-plane or the disk onto any polygonal region — triangles, rectangles, L-shapes, slits, channels with steps.
The idea rests on a simple geometric observation about corners. To map the upper half-plane onto a polygon with interior angles alpha_1 pi, alpha_2 pi, ..., alpha_n pi, you place 'prevertices' x_1 < x_2 < ... < x_n on the real axis (these map to the polygon's corners) and integrate a product of power factors. The formula is f(z) = A + C times the integral from a base point to z of the product over k of (w - x_k)^(alpha_k - 1) dw, where A and C are complex constants fixing position, size, and orientation. Why it works: each factor (w - x_k)^(alpha_k - 1) is holomorphic and nonzero off the real axis, so f is conformal in the half-plane; but as w crosses the real axis at x_k, the argument of that one factor jumps, turning the image direction by exactly the exterior-angle amount (alpha_k - 1) pi — so the image of the real line is a sequence of straight segments meeting at the prescribed angles, i.e. the polygon's boundary. Between the prevertices f' has constant argument, so each piece of the real axis maps to a straight edge.
Schwarz-Christoffel is indispensable in applications: airfoil and channel flows, electrostatic fields in polygonal capacitors, heat flow in regions with corners — all reduce to a half-plane problem once you have the explicit map. Honest caveats, because the formula is genuinely delicate. First, the prevertex positions x_k are NOT free: only the angles are given, but the side LENGTHS depend on the x_k in a transcendental way, so finding the prevertices that produce your specific polygon requires solving a system of equations numerically (the 'parameter problem' or 'crowding'). Second, the integral usually cannot be evaluated in elementary closed form except for special polygons (e.g. a rectangle leads to an elliptic integral). Third, the angle exponents must satisfy the consistency condition that the exterior angles sum to 2 pi (the sum of (1 - alpha_k) over all corners equals 2), reflecting that the boundary turns through one full revolution.
Map the upper half-plane onto an infinite strip 0 < Im W < pi (a 'degenerate polygon' with two corners at infinity). The Schwarz-Christoffel recipe collapses to f(z) = log z: as z runs along the positive real axis the image has constant imaginary part 0, and along the negative real axis it has imaginary part pi, so the two halves of the real line map to the two horizontal edges of the strip.
A degenerate polygon (the strip) recovers the familiar map log z as a Schwarz-Christoffel special case.
Only the corner angles are prescribed by the formula; the prevertex positions (hence side lengths) must usually be found numerically (the parameter problem), and the integral is rarely elementary — a rectangle already requires an elliptic integral.