Mittag-Leffler Function

Author

Eric W. Weisstein
March 10, 2022

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Mittag-LefflerFunction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Mittag-LefflerFunction.html.

©2022 Wolfram Research, Inc. except for portions noted otherwise

Notation

MittagLefflerE[α,x]//TraditionalForm

MittagLefflerE[α,β,x]//TraditionalForm

Sums

(∑_(k=0)^∞)((x^k)/(Γ(k n+1)))==(Subscript[E, n])(x)

$Version

Sum[x^k/Gamma[n k+1],{k,0,Infinity}]

(∑_(k=0)^∞)((x^k)/(Gamma[b+a k]))==(MittagLefflerE2[a, b, x])

Sum[x^k/Gamma[a k+b],{k,0,Infinity}]

Limits

(Limit2Arg[(MittagLefflerE[n, x]), n, 0])==(1/(1-x))

Limit[MittagLefflerE[n,x],n->0]

Special cases

α an integer

$Version

TextGrid[Table[{
n,
HypergeometricPFQ[{},Range[n-1]/n,x/n^n],
MittagLefflerE[n,x]
},{n,0,7}],Dividers->All,Alignment->Left]//TraditionalForm

α a half-integer

$Version

TextGrid[Table[{
n/2,
HypergeometricPFQ[{},Range[n-1]/n,x^2/n^n]+2^((n+1)/2) x HypergeometricPFQ[{1},Range[n+2,3 n,2]/(2 n),x^2/n^n]/n!!/Sqrt[Pi],
MittagLefflerE[n/2,x]
},{n,1,7,2}],Dividers->All,Alignment->Left]//TraditionalForm

Integrals

(∫_0^∞)(e^(-t)) (t^(b-1)) (MittagLefflerE2[a, b, z (t^a)])dt==(1/(1-z))

Integrate[Exp[-t]t^(b-1)MittagLefflerE[a,b,t^a z],{t,0,∞}]//Timing

With[{a=.1,b=2,z=.3},{
NIntegrate[Exp[-t]t^(b-1)MittagLefflerE[a,b,t^a z],{t,0,∞}],
1/(1-z)
}]

Plot

PlotDiscontinuous[Evaluate[Table[MittagLefflerE[n,x],{n,0,5}]],{x,-50,1,10},PlotStyle->{Black,Red,Yellow,Green,Blue,Violet},
AxesLabel->TraditionalForm/@{x,HoldForm[MittagLefflerE[n,x]]}]