Schwarz–Christoffel mapping

In complex analysis, a Schwarz–Christoffel mapping is a conformal transformation of the upper half-plane onto the interior of a simple polygon. Schwarz–Christoffel mappings are used in potential theory and some of its applications, including minimal surfaces and fluid dynamics. They are named after Elwin Bruno Christoffel and Hermann Amandus Schwarz.

Consider a polygon in the complex plane. The Riemann mapping theorem implies that there is a biholomorphic mapping f from the upper half-plane

to the interior of the polygon. The function f maps the real axis to the edges of the polygon. If the polygon has interior angles ${\displaystyle \alpha ,\beta ,\gamma ,\ldots }$, then this mapping is given by

where ${\displaystyle K}$ is a constant, and ${\displaystyle a<b<c<\cdots }$ are the values, along the real axis of the ${\displaystyle \zeta }$ plane, of points corresponding to the vertices of the polygon in the ${\displaystyle z}$ plane. A transformation of this form is called a Schwarz–Christoffel mapping.

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