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.

The Mittag--Leffler function satisfies the recurrence property

$E_{\alpha,\beta}(z)=\frac{1}{z}E_{\alpha,\beta-\alpha}(z)-\frac{1}{z \Gamma(\beta-\alpha),}$

from which the Poincaré asymptotic expansion

$E_{\alpha,\beta}(z)\sim -\sum_{k=1}\frac{1}{z^k \Gamma(\beta-k\alpha)}$

follows, which is true for $z\to-\infty$.

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.

Related to the Laplace transform and Mittag-Leffler summation is the expression

$\int_0^{\infty}e^{-t z} t^{\beta-1} E_{\alpha,\beta}(t^\alpha)dt=\frac{z^{-\beta}}{1-z^{-\alpha}}$

and

$\int_0^\infty e^{-t z} t^{\beta-1} E_{\alpha,\beta}(-t^\alpha) \, dt = \frac{z^{\alpha-\beta}}{1+z^\alpha}$

on the negative axis.

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
• Olver, F. W. J.; Maximon, L. C. (2010), "Mittag-Leffler function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
• Igor Podlubny (1998). "chapter 1". Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering. Academic Press. ISBN 0-12-558840-2.
• Kai Diethelm (2010). "chapter 4". The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type. Lecture notes in mathematics. Heidelberg and New York: Springer-Verlag. ISBN 978-3-642-14573-5.
• Eric W. Weisstein, "Mittag-Leffler function, ", From MathWorld—A Wolfram Web Resource.

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

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