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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Christoffel–Schwarz formula canonical | 1 |
| Schwarz–Christoffel formula | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11812505 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Christoffel–Schwarz formula Context triple: [Elwin Bruno Christoffel, notableWork, Christoffel–Schwarz formula]
-
A.
Bochner–Martinelli formula
The Bochner–Martinelli formula is a fundamental integral representation in several complex variables that generalizes the Cauchy integral formula to higher dimensions.
-
B.
Cauchy–Pompeiu formula
The Cauchy–Pompeiu formula is a fundamental result in complex analysis that extends the Cauchy integral formula to functions that are not necessarily holomorphic by expressing them via both boundary and area integrals.
-
C.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
-
D.
Conformal Invariants
Conformal Invariants is a foundational mathematical work by Lars Ahlfors that systematically develops the theory of quantities preserved under conformal mappings in complex analysis and geometric function theory.
-
E.
Noether’s formula
Noether’s formula is a fundamental result in algebraic geometry that relates the holomorphic Euler characteristic of a smooth projective surface to its Chern numbers, serving as a special case of the Hirzebruch–Riemann–Roch theorem.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Christoffel–Schwarz formula Target entity description: 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.
-
A.
Bochner–Martinelli formula
The Bochner–Martinelli formula is a fundamental integral representation in several complex variables that generalizes the Cauchy integral formula to higher dimensions.
-
B.
Cauchy–Pompeiu formula
The Cauchy–Pompeiu formula is a fundamental result in complex analysis that extends the Cauchy integral formula to functions that are not necessarily holomorphic by expressing them via both boundary and area integrals.
-
C.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
-
D.
Conformal Invariants
Conformal Invariants is a foundational mathematical work by Lars Ahlfors that systematically develops the theory of quantities preserved under conformal mappings in complex analysis and geometric function theory.
-
E.
Noether’s formula
Noether’s formula is a fundamental result in algebraic geometry that relates the holomorphic Euler characteristic of a smooth projective surface to its Chern numbers, serving as a special case of the Hirzebruch–Riemann–Roch theorem.
- F. None of above. chosen
Statements (37)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical formula
ⓘ
result in complex analysis ⓘ |
| alsoKnownAs |
Schwarz–Christoffel formula
ⓘ
linked to:
Christoffel–Schwarz formula
|
| 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
ⓘ
linked to:
Schwarz–Christoffel mapping
|
| 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
ⓘ
Hermann Amandus Schwarz ⓘ |
| relatedTo |
Riemann mapping theorem
ⓘ
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Christoffel–Schwarz formula Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Christoffel–Schwarz formula