---
tag: 12G9M694YG
kind: concept
name: Mittag-Leffler function
msc: ["33E12", "26A33"]
sources:
  - url: https://en.wikipedia.org/wiki/Mittag-Leffler_function
same_as:
  wikidata: Q1935235
  wikipedia: Mittag-Leffler function
---

## Summary

The Mittag-Leffler function $E_{\alpha, \beta}(z) = \sum_{k \ge 0} \frac{z^k}{\Gamma(\alpha k + \beta)}$ generalizes the [[exponential function|0J0FCWV32D]], which is the case $\alpha = \beta = 1$.
It is an [[entire function|0G4F0XB9X2]] for $\operatorname{Re} \alpha > 0$.
It plays the role in [[fractional calculus|T73XK15SA2]] that the exponential plays in ordinary calculus: it solves the basic linear fractional differential equations.

## Explanations

### L1 human

The exponential series is $\sum z^k / k!$, and $k! = \Gamma(k + 1)$.
If $k!$ is replaced by $\Gamma(\alpha k + 1)$, the result is the Mittag-Leffler function $E_\alpha$.
For $\alpha = 1$ it is $e^z$, and for $\alpha = 2$ it is $\cosh \sqrt{z}$.
For $0 < \alpha < 1$, the function $E_\alpha(-t^\alpha)$ decays more slowly than $e^{-t}$, like a power of $t$, which models relaxation in materials with memory.