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:
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.
Sum of a geometric progression:
Mittag-Leffler's integral representation
where the contour C starts and ends at −∞ and circles around the singularities and branch points of the integrand.
- 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.
- Mittag-Leffler function on MathWorld
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