Mittag-Leffler function
E503871
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
complex-valued function
ⓘ
special function ⓘ |
| appearsIn |
fractional-order control systems
ⓘ
solutions of time-fractional diffusion equations ⓘ solutions of time-fractional relaxation equations ⓘ |
| asymptoticBehavior | generalizes exponential-type growth ⓘ |
| conditionOnParameter |
α>0
ⓘ
β>0 ⓘ |
| domain | complex plane ⓘ |
| field |
complex analysis
ⓘ
differential equations ⓘ fractional calculus ⓘ integral equations ⓘ |
| generalizes | exponential function ⓘ |
| growthType | order 1/α entire function ⓘ |
| hasGeneralForm | E_{α,β}(z) NERFINISHED ⓘ |
| hasSpecialCase | E_{α}(z) NERFINISHED ⓘ |
| hasVariant |
three-parameter Mittag-Leffler function
ⓘ
two-parameter Mittag-Leffler function ⓘ |
| introducedBy | Gösta Mittag-Leffler NERFINISHED ⓘ |
| introducedIn | early 20th century ⓘ |
| isEntireFunction | true ⓘ |
| namedAfter | Gösta Mittag-Leffler NERFINISHED ⓘ |
| parameter |
α
ⓘ
β ⓘ |
| property |
completely monotone on (0,∞) for certain parameter ranges
ⓘ
interpolates between power-law and exponential behavior ⓘ |
| relatedTo |
Gamma function
NERFINISHED
ⓘ
Laplace transform NERFINISHED ⓘ fractional derivative ⓘ fractional integral ⓘ |
| role | fundamental solution of many fractional differential equations ⓘ |
| seriesDefinition |
E_{α,β}(z)=∑_{k=0}^{∞} z^{k}/Γ(α k+β)
ⓘ
E_{α}(z)=∑_{k=0}^{∞} z^{k}/Γ(α k+1) ⓘ |
| specialCase |
E_{1,1}(z)=e^{z}
ⓘ
E_{1}(z)=e^{z} ⓘ E_{α,1}(z)=E_{α}(z) ⓘ |
| threeParameterNotation | E_{α,β}^{γ}(z) ⓘ |
| usedIn |
anomalous diffusion models
ⓘ
control theory ⓘ fractional differential equations ⓘ fractional integral equations ⓘ probability theory ⓘ viscoelasticity ⓘ |
| usedToModel |
memory effects in complex media
ⓘ
relaxation processes ⓘ subdiffusion ⓘ superdiffusion ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.