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Mittag-Leffler 函数

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MittagLeffler

Mittag-Leffler 函数（Mittag-Leffler 1903, 1905）是由级数定义的整函数

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

对于 alpha>0。它与广义双曲函数 F_(alpha,r)^alpha(z) 相关，关系为

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

并用广义合流超几何函数显式地给出为

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

它在 Wolfram 语言 中实现为MittagLefflerE[a, z] 和MittagLefflerE[a, b, z]。

Mittag-Leffler 函数自然地出现在分数阶积分方程（Saxena et al. 2002）的解中，尤其是在动力学方程、随机游走、莱维飞行和所谓的超扩散输运的分数阶推广的研究中。普通和广义 Mittag-Leffler 函数在纯指数定律和由普通动力学方程及其分数阶对应物控制的现象的幂律行为之间进行插值（Lang 1999ab, Hilfer 2000, Saxena et al. 2002）。

整数 alpha=n 的特殊值是

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)

对于半整数 n/2，这些函数可以显式地写为

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)

给出特殊值

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

对于 z>0，其中 erf(z) 是 erf，erfc(z) 是 erfc（Saxena et al. 2002）。可以看出，E_(1/2)(z) 与 道森积分 D_-(z) 密切相关。

更一般的 Mittag-Leffler 函数

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

也可以为 alpha,beta>0 定义（Wiman 1905, Agarwal 1953, Humbert 1953, Humbert and Agarwal 1953, Gorenflo 1987, Miller 1993, Mainardi and Gorenflo 1996, Gorenflo 1998, Sixdeniers et al. 1999），因此

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

一般的 Mittag-Leffler 函数可以用 Fox H-函数 表示（Saxena et al. 2002）。

一般的 Mittag-Leffler 函数满足

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

对于 |z|<1 （Erdélyi et al. 1981, p. 210; Samko et al. 1993, p. 21），这给出了 E_(alpha,beta)(z) 的 拉普拉斯变换 为

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

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

另请参阅

道森积分, 广义双曲函数, Wright 函数

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参考文献

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 函数

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Weisstein, Eric W. "Mittag-Leffler 函数。" 来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/Mittag-LefflerFunction.html

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