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Mittag-Leffler Function

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MittagLeffler

The Mittag-Leffler function (Mittag-Leffler 1903, 1905) is an entire function defined by the series

E_alpha(z)=sum_(k=0)^infty(z^k)/(Gamma(alphak+1))
(1)

for alpha>0. It is related to the generalized hyperbolic functions F_(alpha,r)^alpha(z) by

F_(alpha,0)^1(z)=E_alpha(z^n),
(2)

and given explicitly in terms of generalized confluent hypergeometric functions as

E_alpha(z)=_0F_(alpha-1)(;1/alpha,2/alpha,...,(alpha-1)/alpha;z/(alpha^alpha)).
(3)

It is implemented in the Wolfram Language as MittagLefflerE[a, z] and MittagLefflerE[a, b, z].

The Mittag-Leffler function arises naturally in the solution of fractional integral equations (Saxena et al. 2002), and especially in the study of the fractional generalization of the kinetic equation, random walks, Lévy flights, and so-called superdiffusive transport. The ordinary and generalized Mittag-Leffler functions interpolate between a purely exponential law and power-like behavior of phenomena governed by ordinary kinetic equations and their fractional counterparts (Lang 1999ab, Hilfer 2000, Saxena et al. 2002).

Special values for integer alpha=n are

E_0(z)=1/(1-z)
(4)
E_1(z)=e^z
(5)
E_2(z)=cosh(sqrt(z))
(6)
E_3(z)=1/3[e^(z^(1/3))+2e^(-z^(1/3)/2)cos(1/2sqrt(3)z^(1/3))]
(7)
E_4(z)=1/2[cos(z^(1/4))+cosh(z^(1/4))].
(8)

For half-integers n/2, the functions can be written explicitly as

E_(n/2)(z)=_0F_(n-1)(;1/n,2/n,...,(n-1)/n;(z^2)/(n^n))+(2^((n+1)/2)z)/(n!!sqrt(pi))_1F_alpha(1;(n+2)/(2n),(n+3)/(2n),...,(3n)/(2n);(z^2)/(n^n)),
(9)

giving the special value

E_(1/2)(z)=e^(z^2)[1+erf(z)]
(10)
=e^(z^2)erfc(-z),
(11)

for z>0, where erf(z) is erf and erfc(z) is erfc (Saxena et al. 2002). As can be seen, E_(1/2)(z) is closely related to Dawson's integral D_-(z).

The more general Mittag-Leffler function

E_(alpha,beta)(z)=sum_(k=0)^infty(z^k)/(Gamma(alphak+beta))
(12)

can also be defined for alpha,beta>0 (Wiman 1905, Agarwal 1953, Humbert 1953, Humbert and Agarwal 1953, Gorenflo 1987, Miller 1993, Mainardi and Gorenflo 1996, Gorenflo 1998, Sixdeniers et al. 1999), so that

E_alpha(z)=E_(alpha,1)(z).
(13)

The general Mittag-Leffler function can be representation in terms of Fox H-functions (Saxena et al. 2002).

The general Mittag-Leffler function satisfies

int_0^inftye^(-t)t^(beta-1)E_(alpha,beta)(t^alphaz)dt=1/(1-x)
(14)

for |z|<1 (Erdélyi et al. 1981, p. 210; Samko et al. 1993, p. 21), which gives the Laplace transform of E_(alpha,beta)(z) as

int_0^inftye^(pt)t^(beta-1)E_(alpha,beta)(at^alpha)dt=p^(-beta)(1-ap^(-alpha))^(-1)
(15)

for R[p]>|a|^(1/alpha) and R[beta]>0 (Samko 1993, p. 21; Saxena et al. 2002).

See also

Dawson's Integral, Generalized Hyperbolic Functions, Wright Function

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References

Agarwal, R. P. "A propos d'une note de M. Pierre Humbert." Comptes Rendus Acad. Sci. Paris 236, 2031-2032, 1953.Dzherbashyan, M. M. Integral Transform Representations of Functions in the Complex Domain. Moscow: Nauka, 1966.Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "Mittag-Leffler's Function E_alpha(z) and Related Functions." §18.1 in Higher Transcendental Functions, Vol. 3. New York: Krieger, pp. 206-212, 1981.Gorenflo, R. "Newtonsche Aufheizung, Abelsche Integralgleichungen zweiter Art und Mittag-Leffler-Funktionen." Z. Naturforsch. A 42, 1141-1146, 1987.Gorenflo, R.; Kilbas, A. A.; and Rogosin, S. V. "On the Generalized Mittag-Leffler Type Functions." Integral Transform. Spec. Funct. 7, 215-224, 1998.Hilfer, R. and Anton, L. "Fractional Master Equations and Fractal Time Random Walks." Phys. Rev. E 51, R848-R851, 1995.Hilfer, R. "On Fractional Diffusion and Its Relation with Continuous Time Random Walks." In Anomalous Diffusion: From Basics to Application: Proceedings of the XIth Max Born Symposium Held at Ladek Zdroj, Poland, 20-27 May 1998 (Ed. R. Kutner, A. Pekalski, and K. Sznaij-Weron). Berlin: Springer-Verlag, pp. 77-82, 1999.Hilfer, R. (Ed.). Applications of Fractional Calculus in Physics. Singapore: World Scientific, 2000.Humbert, P. "Quelques résultats relatifs à la fonction de Mittag-Leffler." Comptes Rendus Acad. Sci. Paris 236, 1467-1468, 1953.Humbert, P. and Agarwal, R. P. "Sur la fonction de Mittag-Leffler et quelques-unes de ses généralisations." Bull. Sci. Math. Ser. 2 77, 180-185, 1953.Humbert, P. and Delerue, P. "Sur une extension à deux variables de la fonction de Mittag-Leffler." Comptes Rendus Acad. Sci. Paris 237, 1059-1060, 1953.Lang, K. R. Astrophysical Formulae, Vol. 1: Radiation, Gas Processes, and High-Energy Astrophysics, 3rd enl. rev. ed. New York: Springer-Verlag, 1999a.Lang, K. R. Astrophysical Formulae, Vol. 2: Space, Time, Matter and Cosmology. New York: Springer-Verlag, 1999b.Mainardi, F. and Gorenflo, R. "The Mittag-Leffler Function in the Riemann-Liouville Fractional Calculus." In Proceedings of the International Conference Dedicated to the Memory of Academician F. D. Gakhov; Held in Minsk, February 16-20, 1996 (Ed. A. A. Kilbas). Minsk, Beloruss: Beloruss. Gos. Univ., Minsk, pp. 215-225, 1996.Meerschaert, M. M.; Benson, D. A.; Scheffler, H.-P.; and Baeumer, B. "Stochastic Solution of Space-Time Fractional Diffusion Equations." Phys. Rev. E 65, 041103, 2002.Miller, K. S. "The Mittag-Leffler and Related Functions." Integral Transform. Spec. Funct. 1, 41-49, 1993.Mittag-Leffler, M. G. "Sur la nouvelle fonction E_alpha(x)." Comptes Rendus Acad. Sci. Paris 137, 554-558, 1903.Mittag-Leffler, M. G. "Sur la representation analytique d'une branche uniforme d'une fonction monogene." Acta Math. 29, 101-181, 1905.Muldoon, M. E. and Ungar, A. A. "Beyond Sin and Cos." Math. Mag. 69, 3-14, 1996.Podlubny, I. "The Laplace Transform Method for Linear Differential equations of the Fractional Order." 30 Oct 1997. http://arxiv.org/abs/funct-an/9710005.Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, pp. 21-22, 1993.Saxena, R. K.; Mathai, A. M.; and Haubold, H. J. "On Fractional Kinetic Equations." 23 Jun 2002. http://arxiv.org/abs/math.CA/0206240.Sixdeniers, J.-M.; Penson, K. A.; and Solomon, A. I. "Mittag-Leffler Coherent States." J. Phys. A: Math. Gen. 32, 7543-7563, 1999.Sokolov, I. M.; Klafter, J. and Blumen, A. "Do Strange Kinetics Imply Unusual Thermodynamics?" Phys. Rev. E 64, 021107, 2001.Wiman, A. "Über den Fundamentalsatz in der Theorie der Funktionen E_a(x)." Acta Math. 29, 191-201, 1905.

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Mittag-Leffler Function

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Weisstein, Eric W. "Mittag-Leffler Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Mittag-LefflerFunction.html

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