MittagLefflerE

MittagLefflerE[α,z]

gives the Mittag–Leffler function .

MittagLefflerE[α,β,z]

gives the generalized Mittag–Leffler function .

Details

MittagLefflerE is a mathematical function, suitable for both symbolic and numerical manipulation.
MittagLefflerE is typically used in the solution of fractional-order differential equations, similar to the Exp function in the solution of ordinary differential equations.
MittagLefflerE allows to be any non-negative real number.
The generalized Mittag–Leffler function is an entire function of given by its defining series $E_(alpha,beta)(z)=sum_(k=0)^inftyz^k/TemplateBox[{{{alpha, , k}, +, beta}}, Gamma]$ .
The Mittag–Leffler function is equivalent to .

Examples

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Basic Examples (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope (28)

Numerical Evaluation (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

Specific Values (4)

Simple exact values are generated automatically:

Evaluate symbolically:

Values at infinity:

Find a value of x for which MittagLefflerE[1/2,x]=0.5:

Visualization (3)

Plot the MittagLefflerE function for noninteger orders:

Plot MittagLefflerE function for integer orders:

Plot the real part of :

Plot the imaginary part of :

Function Properties (9)

is defined for all as long as :

The complex domain of MittagLefflerE is the same:

MittagLefflerE has the mirror property :

MittagLefflerE threads elementwise over lists:

MittagLefflerE is an analytic function for :

It is singular and discontinuous for :

is injective:

is not surjective:

is non-negative:

is simplified to :

and are simplified to elementary functions:

Differentiation (2)

First derivatives with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z when a=1/4:

Series Expansions (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

Fractional Differential Equations (3)

MittagLefflerE plays an important role in solutions of fractional DEs with constant coefficients:

Verify it:

Plot this solution:

Solve a fractional DE with constant coefficients containing two different order Caputo derivatives:

Solve a system of two fractional DEs in vector form:

Plot the solution:

Parametrically plot the solution:

Integral Transforms (1)

Laplace transform of specific MittagLefflerE functions:

ComplexPlot in the -domain:

Apply InverseLaplaceTransform to transform back to the time domain and get the initial expression:

Applications (3)

Define a Mittag–Leffler random variate for :

A Mittag–Leffler random variate is related to the positive stable random variate:

Generate random variates and compare the histogram to the distribution density:

InverseLaplaceTransform of this algebraic function is given in terms of MittagLefflerE:

The family of MittagLefflerE functions is FoxH representable:

Properties & Relations (3)

The Mittag–Leffler function is closed under differentiation:

The function is simplified to HypergeometricPFQ for non-negative integer :

While for non-negative half-integer it is simplified to the sum of HypergeometricPFQ:

The Mittag–Leffler function is an infinite sum:

For specific values of , this sum might be written in terms of HypergeometricPFQ functions:

Compare this with the MittagLefflerE output:

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