Fuchsian singularity
E500440
A Fuchsian singularity is a type of regular singular point of a linear differential equation in the complex plane, characterized by well-controlled (typically polynomially bounded) behavior of solutions near the singularity.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
type of singularity of differential equation ⓘ |
| appliesTo |
linear differential equation
ⓘ
linear ordinary differential equation ⓘ |
| belongsTo | singularities of linear differential equations on Riemann surfaces ⓘ |
| characterizedBy |
polynomially bounded growth of solutions near the singularity
ⓘ
regular singular behavior at a point ⓘ well-controlled behavior of solutions near the singularity ⓘ |
| classificationCriterion |
analyticity of suitably rescaled coefficient functions at the singular point
ⓘ
growth rate of fundamental solutions near the singular point ⓘ |
| conditionOnCoefficients |
(z - z0) a_k(z) / a_n(z) is analytic at z0 for k < n
ⓘ
(z - z0)^{n-k} a_k(z) / a_n(z) is analytic at z0 for k < n ⓘ |
| contrastsWith |
essential singularity of differential equation
ⓘ
irregular singularity ⓘ |
| domain | complex plane ⓘ |
| field |
complex analysis
ⓘ
differential equations in the complex plane ⓘ ordinary differential equations ⓘ |
| generalizationOf | regular singular point on the Riemann sphere ⓘ |
| hasExample |
Bessel equation singularity at zero
ⓘ
hypergeometric equation singularities at 0, 1, and infinity ⓘ |
| hasOppositePropertyTo | irregular singularity with essential exponential growth ⓘ |
| hasProperty |
solutions can be expressed using Frobenius series
ⓘ
solutions have at most polynomial growth in a punctured neighborhood ⓘ |
| hasTerminologyVariant |
Fuchsian singular point
ⓘ
regular singularity of Fuchs type ⓘ |
| implies |
local solutions have moderate growth near z0
ⓘ
no essential exponential blow-up of solutions at the singularity ⓘ |
| isA | regular singular point ⓘ |
| localBehavior |
solutions are meromorphic after suitable ramified covering
ⓘ
solutions may involve powers and logarithms of (z - z0) ⓘ |
| namedAfter | Lazarus Fuchs NERFINISHED ⓘ |
| occursAt | singular point z0 of a linear differential equation ⓘ |
| occursIn | Fuchsian system of linear differential equations ⓘ |
| relatedConcept |
Frobenius method
NERFINISHED
ⓘ
Fuchsian differential equation NERFINISHED ⓘ indicial equation ⓘ monodromy of differential equation ⓘ regular singular point ⓘ |
| relatedTo | Fuchsian group (via historical naming, not direct equivalence) ⓘ |
| studiedBy | Lazarus Fuchs NERFINISHED ⓘ |
| studiedIn | 19th-century theory of linear differential equations ⓘ |
| usedIn |
Riemann–Hilbert correspondence
NERFINISHED
ⓘ
classification of singularities of linear differential equations ⓘ study of monodromy representations ⓘ theory of linear ODEs with singular points ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.