page_title: Schwarz–Christoffel mapping
section_title: Example
hierarchical_section_title: Schwarz–Christoffel mapping / Example
caption_attribution_description: English: Schwarz-Christoffel mapping of the semi-infinite strip to the upper half-plane.
context_page_description: 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.
context_section_description: 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
  Evaluation of this integral yields
  where C is a (complex) constant of integration.  Requiring that f(−1) = Q and f(1) = P gives C = 0 and K = 1. Hence the Schwarz–Christoffel mapping is given by
  This transformation is sketched below.
caption: Schwarz–Christoffel mapping of the upper half-plane to the semi-infinite strip
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