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)
Series expansion at Infinity:
Numerical Evaluation (4)
Specific Values (4)
Find a value of x for which MittagLefflerE[1/2,x]=0.5:
Function Properties (9)
The complex domain of MittagLefflerE is the same:
MittagLefflerE has the mirror property :
MittagLefflerE threads elementwise over lists:
MittagLefflerE is an analytic function for :
Series Expansions (2)
Find the Taylor expansion using Series:
Fractional Differential Equations (3)
MittagLefflerE plays an important role in solutions of fractional DEs with constant coefficients:
Properties & Relations (3)
The function is simplified to HypergeometricPFQ for non-negative integer :
While for non-negative half-integer it is simplified to the sum of HypergeometricPFQ:
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