Christoffel–Schwarz formula
E947535
The Christoffel–Schwarz formula is a fundamental result in complex analysis that provides an explicit conformal mapping from the upper half-plane onto polygonal regions in the complex plane.
Statements (37)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical formula
ⓘ
result in complex analysis ⓘ |
| alsoKnownAs | Schwarz–Christoffel formula NERFINISHED ⓘ |
| appearsIn |
advanced textbooks on complex analysis
ⓘ
literature on numerical conformal mapping algorithms ⓘ |
| appliesTo | simply connected polygonal regions ⓘ |
| assumes |
angles of polygon are less than 2π
ⓘ
polygon has finitely many vertices ⓘ |
| category | conformal mapping theory ⓘ |
| codomain | polygonal region in the complex plane ⓘ |
| describes | conformal mapping from upper half-plane to polygonal regions ⓘ |
| domain | upper half-plane ⓘ |
| field | complex analysis ⓘ |
| gives | explicit formula for conformal maps onto polygons ⓘ |
| hasGeneralization | Schwarz–Christoffel mapping for the unit disk NERFINISHED ⓘ |
| hasParameter |
additive complex constant
ⓘ
locations of prevertices on the real axis ⓘ multiplicative complex constant ⓘ |
| implies | conformal equivalence between upper half-plane and any simply connected polygonal region (except whole plane) ⓘ |
| involves | integral of a product of powers of linear factors ⓘ |
| isToolIn |
geometric function theory
ⓘ
numerical conformal mapping ⓘ |
| maps | real axis to boundary of a polygon ⓘ |
| namedAfter |
Elwin Bruno Christoffel
NERFINISHED
ⓘ
Hermann Amandus Schwarz NERFINISHED ⓘ |
| relatedTo |
Riemann mapping theorem
NERFINISHED
ⓘ
conformal equivalence of simply connected domains ⓘ |
| relates | prevertices on the real axis to vertices of a polygon ⓘ |
| requires |
choice of prevertices on the real line
ⓘ
interior angles of the target polygon ⓘ |
| typeOf | integral representation of conformal maps ⓘ |
| usedFor |
constructing conformal maps onto polygonal domains
ⓘ
solving boundary value problems via conformal mapping ⓘ |
| usedIn |
aerodynamics
ⓘ
electrostatics ⓘ engineering applications of potential theory ⓘ fluid dynamics ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.