# MittagLefflerE

MittagLefflerE[α,z]

gives the MittagLeffler function .

MittagLefflerE[α,β,z]

gives the generalized MittagLeffler function .

# Details • MittagLefflerE is a mathematical function, suitable for both symbolic and numerical manipulation.
• MittagLefflerE allows to be any positive real number.
• Generalized MittagLeffler function is an entire function of given by its defining series .
• MittagLeffler 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(19)

### 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(4)

Real domain of MittagLefflerE:

Complex domain of MittagLefflerE:

The function range of MittagLefflerE:

The range for complex values:

MittagLefflerE has the mirror property :

### 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 MittagLeffler random variate for :

A MittagLeffler 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 MittagLeffler function is closed under differentiation:

The MittagLeffler function for integer parameter satisfies a simple differential equation:

Verify for several integer parameters:

Introduced in 2012
(9.0)