gives the Mittag–Leffler function .
gives the generalized Mittag–Leffler function .
- 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 .
- The Mittag–Leffler function is equivalent to .
Examplesopen allclose all
Basic Examples (5)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Numerical Evaluation (4)
Specific Values (4)
Simple exact values are generated automatically:
Find a value of x for which MittagLefflerE[1/2,x]=0.5:
Plot the MittagLefflerE function for noninteger orders:
Plot MittagLefflerE function for integer orders:
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 :
Series Expansions (2)
Find the Taylor expansion using Series:
Plots of the first three approximations around :
Fractional Differential Equations (3)
MittagLefflerE plays an important role in solutions of fractional DEs with constant coefficients:
Solve a fractional DE with constant coefficients containing two different order Caputo derivatives:
Solve a system of two fractional DEs in vector form:
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:
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:
Wolfram Research (2012), MittagLefflerE, Wolfram Language function, https://reference.wolfram.com/language/ref/MittagLefflerE.html (updated 2022).
Wolfram Language. 2012. "MittagLefflerE." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/MittagLefflerE.html.
Wolfram Language. (2012). MittagLefflerE. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MittagLefflerE.html