5.4 The Schwarz–Christoffel Transformation

The Riemann Mapping Theorem is elegant in its own simplicity and definitions. However, it is only a theorem that guarantees existence of biholomorphisms. No information whatsoever can be straightforwardly extracted regarding the explicit construction of such biholomorphisms. However, in the explicit case that is the open interior of a polygon, the result is provided by the Schwarz–Christoffel Transformation.

Let be distinct real numbers. Suppose are positive real numbers satisfying . Let

where the branch of each factor is selected to be

where the branch of is selected such that , holomorphic on (the lower imaginary axis is known as a branch cut). For , the argument of this factor is . For ,

achieved by selecting branches of each factor by the method described earlier.

Let be fixed. If , the branches of all where have vanishing arguments; hence,

If , we have

Therefore, we can define complex numbers via

where is fixed.

The absolute integrability of along the real axis concerns only the convergence at each singularity and the behavior as . For a fixed , (where is holomorphic and nonzero in a compact neighborhood of ). Since , it is an integrable singularity. Since as and , is integrable on .

Let

Since is holomorphic on ,