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 this case, 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.

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

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:

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

Sum of a geometric progression:

$E_{0,1}(z) = \frac{1}{1-z}.$

Hyperbolic cosine:

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

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.

References

• Olver, F. W. J.; Maximon, L. C. (2010), "Mittag-Leffler function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248, http://dlmf.nist.gov/10.46 
• 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, vol. 198), by Igor Podlubny. Hardcover. Publisher: Academic Press; (October 1998). ISBN 0-12-558840-2 (see Chapter 1).

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

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