Fuchsian differential equation
E500438
A Fuchsian differential equation is a type of linear ordinary differential equation characterized by having only regular singular points, extensively studied in complex analysis and the theory of special functions.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Legendre’s differential equation | 1 |
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
differential equation
ⓘ
linear ordinary differential equation ⓘ mathematical object ⓘ |
| appearsIn |
algebraic geometry via Picard–Fuchs equations
ⓘ
isomonodromic deformation theory ⓘ theory of modular forms ⓘ |
| characterizedBy | having only regular singular points ⓘ |
| classificationBy | position and nature of singular points ⓘ |
| domain | complex plane ⓘ |
| excludesSingularities | irregular singular points ⓘ |
| field |
complex analysis
ⓘ
ordinary differential equations ⓘ theory of special functions ⓘ |
| generalizes | hypergeometric equation ⓘ |
| governs | analytic continuation of solutions around singularities ⓘ |
| hasApplication |
deformation of complex structures
ⓘ
integrable systems ⓘ mathematical physics ⓘ quantum field theory ⓘ |
| hasCondition |
coefficients have at most simple poles at singular points
ⓘ
solutions grow at most polynomially near singular points ⓘ |
| hasInvariant |
local exponents at singular points
ⓘ
monodromy representation ⓘ |
| hasProperty |
linear
ⓘ
second-order in many classical examples ⓘ |
| hasSingularities | regular singular points ⓘ |
| hasType |
matrix Fuchsian differential equation
ⓘ
scalar Fuchsian differential equation ⓘ |
| localBehaviorDescribedBy | Frobenius method ⓘ |
| namedAfter | Lazarus Fuchs NERFINISHED ⓘ |
| oftenConsideredOn | Riemann sphere NERFINISHED ⓘ |
| relatedConcept |
Fuchs’ theorem
NERFINISHED
ⓘ
irregular singular point ⓘ regular singular point ⓘ |
| relatedTo |
Fuchsian group
NERFINISHED
ⓘ
Gauss hypergeometric function NERFINISHED ⓘ Heun equation ⓘ Picard–Fuchs equation NERFINISHED ⓘ Riemann–Hilbert problem NERFINISHED ⓘ hypergeometric differential equation ⓘ |
| solutionSpace | space of analytic functions with controlled growth near singularities ⓘ |
| studiedInContextOf |
Riemann surfaces
NERFINISHED
ⓘ
analytic continuation ⓘ monodromy theory ⓘ |
| studiedSince | 19th century ⓘ |
| usedToDefine | special functions ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Legendre’s differential equation