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 allows to be any positive 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 (22)

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 (7)

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:

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:

Applications (1)

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:

Properties & Relations (2)

The Mittag–Leffler function is closed under differentiation:

The Mittag–Leffler function for integer parameter satisfies a simple differential equation:

Verify for several integer parameters:

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