Mittag-Leffler function

In mathematics, the Mittag-Leffler function E_α,β is a special function, a complex function which depends on two complex parameters α and β. It may be defined by the following series when the real part of α is strictly positive:

$E_{\alpha, \beta} (z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma(\alpha k + \beta)}.$

In the case α and β are real and positive, the series converges for all values of the argument z, so the Mittag-Leffler function is an entire function. This function is named after Gösta Mittag-Leffler. This class of functions are important in the theory of the fractional calculus.

For α > 0, the Mittag-Leffler function E_α,1 is an entire function of order 1/α, and is in some sense the simplest entire function of its order.

Special cases

For $\alpha=0,1/2,1,2$ we find

The sum of a geometric progression:

$E_{0,1}(z) = \sum_{k=0}^\infty z^k = \frac{1}{1-z}.$

Exponential function:

$E_{1,1}(z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma (k + 1)} = \sum_{k=0}^\infty \frac{z^k}{k!} = \exp(z).$

Error function:

$E_{1/2,1}(z) = \exp(z^2)\operatorname{erfc}(-z).$

Hyperbolic cosine:

$E_{2,1}(z) = \cosh(\sqrt{z}).$

For $\alpha=0,1,2$, the integral

$\int_0^{z}E_{\alpha,1}(-s^2){\mathrm d}s$

gives, respectively

$\arctan(z)$,

$\tfrac{\sqrt{\pi}}{2}\operatorname{erf}(z)$,

$\sin(z)$.

Mittag-Leffler's integral representation

$E_{\alpha,\beta}(z)=\frac{1}{2\pi i}\int_C \frac{t^{\alpha-\beta}e^t}{t^\alpha-z} \, dt$

where the contour C starts and ends at −∞ and circles around the singularities and branch points of the integrand.

See also

• Mittag-Leffler summation
• Mittag-Leffler distribution
• Fox–Wright function

References

• Mittag-Leffler, M.G.: Sur la nouvelle fonction E(x). C. R. Acad. Sci. Paris 137, 554–558 (1903)
• Mittag-Leffler, M.G.: Sopra la funzione E˛.x/. Rend. R. Acc. Lincei, (Ser. 5) 13, 3–5 (1904)
• Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V., Mittag-Leffler Functions, Related Topics and Applications (Springer, New York, 2014) 443 pages ISBN 978-3-662-43929-6

External links

• Mittag-Leffler function on MathWorld
• Mittag-Leffler function: MATLAB code
• Mittag-Leffler and stable random numbers: Continuous-time random walks and stochastic solution of space-time fractional diffusion equations
• Parameter estimation for fractional Poisson processes

This article incorporates material from Mittag-Leffler function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.