# 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 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 MittagLeffler function is an entire function of given by its defining series .
• The 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(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 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 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:

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

The family of MittagLefflerE functions is FoxH representable:

## Properties & Relations(3)

The MittagLeffler 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 MittagLeffler 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: