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 alpha to be any non-negative real number.
  • The generalized MittagLeffler 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 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 TemplateBox[{2, {x, +, {i,  , y}}}, MittagLefflerE]:

Plot the imaginary part of TemplateBox[{2, {x, +, {i,  , y}}}, MittagLefflerE]:

Function Properties  (9)

TemplateBox[{a, x}, MittagLefflerE] is defined for all as long as :

The complex domain of MittagLefflerE is the same:

MittagLefflerE has the mirror property TemplateBox[{1, {z, }}, MittagLefflerE]=TemplateBox[{1, z}, MittagLefflerE]:

MittagLefflerE threads elementwise over lists:

MittagLefflerE is an analytic function for :

It is singular and discontinuous for :

TemplateBox[{2, x}, MittagLefflerE] is injective:

TemplateBox[{{1, /, 2}, x}, MittagLefflerE] is not surjective:

TemplateBox[{{1, /, 2}, x}, MittagLefflerE] is non-negative:

TemplateBox[{{a, ,, 1}, x}, MittagLefflerE] is simplified to TemplateBox[{a, x}, MittagLefflerE]:

TemplateBox[{{0, ,, 1}, x}, MittagLefflerE] and TemplateBox[{0, x}, MittagLefflerE] 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  (2)

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:

Properties & Relations  (3)

The MittagLeffler function is closed under differentiation:

The TemplateBox[{a, x}, MittagLefflerE] 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:

Wolfram Research (2012), MittagLefflerE, Wolfram Language function, https://reference.wolfram.com/language/ref/MittagLefflerE.html (updated 2022).

Text

Wolfram Research (2012), MittagLefflerE, Wolfram Language function, https://reference.wolfram.com/language/ref/MittagLefflerE.html (updated 2022).

CMS

Wolfram Language. 2012. "MittagLefflerE." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/MittagLefflerE.html.

APA

Wolfram Language. (2012). MittagLefflerE. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MittagLefflerE.html

BibTeX

@misc{reference.wolfram_2022_mittaglefflere, author="Wolfram Research", title="{MittagLefflerE}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/MittagLefflerE.html}", note=[Accessed: 28-June-2022 ]}

BibLaTeX

@online{reference.wolfram_2022_mittaglefflere, organization={Wolfram Research}, title={MittagLefflerE}, year={2022}, url={https://reference.wolfram.com/language/ref/MittagLefflerE.html}, note=[Accessed: 28-June-2022 ]}