Mittag-Leffler function

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In mathematics, the Mittag-Leffler function Eα,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 Eα,1 is an entire function of order 1/α, and is in some sense the simplest entire function of its order.

Special cases[edit]

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[edit]

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[edit]

References[edit]

External links[edit]

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