Schwarz–Christoffel mapping
E947534
The Schwarz–Christoffel mapping is a conformal transformation that maps the upper half-plane (or unit disk) onto polygonal regions, playing a central role in complex analysis and applications such as fluid dynamics and electrostatics.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
conformal mapping
ⓘ
mathematical concept ⓘ tool in complex analysis ⓘ |
| assumes |
target boundary is a polygon or polygonal arc
ⓘ
target domain is simply connected ⓘ |
| belongsTo |
geometric function theory
ⓘ
theory of univalent functions ⓘ |
| computationalAspect |
often implemented via numerical methods
ⓘ
requires solving nonlinear systems for prevertices ⓘ |
| condition |
sum of (1−α_k)=2 for finite polygon
ⓘ
sum of turning angles equals 2π ⓘ |
| difficulty |
evaluation of multivalued integrals
ⓘ
parameter problem for locating prevertices ⓘ |
| field | complex analysis ⓘ |
| generalizedBy |
Schwarz–Christoffel mappings for multiply connected domains
NERFINISHED
ⓘ
Schwarz–Christoffel mappings for regions with slits ⓘ |
| hasApplication |
design of channels and nozzles
ⓘ
electrostatic field computation near polygonal conductors ⓘ mesh generation in computational physics ⓘ modeling flow past obstacles ⓘ |
| hasBoundaryCorrespondence |
real axis to polygon boundary
ⓘ
unit circle to polygon boundary ⓘ |
| hasFormula | f(z)=A∫∏(ζ−z_k)^{α_k−1} dζ + B, where α_k are interior angle parameters ⓘ |
| hasVariant |
disk-to-polygon Schwarz–Christoffel formula
NERFINISHED
ⓘ
half-plane-to-polygon Schwarz–Christoffel formula ⓘ |
| historicalPeriod | 19th century mathematics ⓘ |
| mapsFrom |
unit disk
ⓘ
upper half-plane ⓘ |
| mapsTo |
polygonal region
ⓘ
simply connected polygonal domain ⓘ |
| namedAfter |
Elwin Bruno Christoffel
NERFINISHED
ⓘ
Hermann Amandus Schwarz NERFINISHED ⓘ |
| parameter |
complex constants A and B
ⓘ
interior angles of polygon ⓘ prevertices on the real axis ⓘ |
| property |
angle-preserving
ⓘ
extends continuously to boundary almost everywhere ⓘ holomorphic on the upper half-plane except at prevertices ⓘ |
| relatedTo |
Riemann mapping theorem
NERFINISHED
ⓘ
conformal equivalence of simply connected domains ⓘ numerical conformal mapping ⓘ |
| requires | choice of branch cuts for fractional powers ⓘ |
| usedFor |
aerodynamics
ⓘ
electrostatics ⓘ fluid dynamics ⓘ free-boundary problems ⓘ heat conduction problems ⓘ mapping canonical domains to polygonal regions ⓘ solving boundary value problems ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.