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. 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 α , β , γ , … {displaystyle alpha ,eta ,gamma ,ldots } , then this mapping is given by where K {displaystyle K} is a constant, and a < b < c < ⋯ {displaystyle a<b<c<cdots } are the values, along the real axis of the ζ {displaystyle zeta } plane, of points corresponding to the vertices of the polygon in the z {displaystyle z} plane. A transformation of this form is called a Schwarz–Christoffel mapping. The integral can be simplified by mapping the point at infinity of the ζ {displaystyle zeta } plane to one of the vertices of the z {displaystyle z} plane polygon. By doing this, the first factor in the formula becomes constant and so can be absorbed into the constant K {displaystyle K} . Conventionally, the point at infinity would be mapped to the vertex with angle α {displaystyle alpha } . Consider a semi-infinite strip in the z plane. This may be regarded as a limiting form of a triangle with vertices P = 0, Q = π i, and R (with R real), as R tends to infinity. Now α = 0 and β = γ = ​π⁄2 in the limit. Suppose we are looking for the mapping f with f(−1) = Q, f(1) = P, and f(∞) = R. Then f is given by