米塔-列夫勒函数

米塔-列夫勒函数(Mittag-Leffler function)是一个特殊函数，常用于分数微积分方程，定义如下

Mittag-Leffler function 2D plot

Mittag-Leffler function Imaginary 3D plot1

Mittag-Leffler function Imaginary 3D plot2

$E_{a,b}(z)=\sum _{k=0}^{\infty }{\frac {z^{k}}{\Gamma (ak+b)}}$

特例

对应 $a=0,1/2,1,2$ 有

$E_{0,1}(z)=\sum _{k=0}^{\infty }z^{k}={\frac {1}{1-z}}.$

指数函数:

$E_{1,1}(z)=\sum _{k=0}^{\infty }{\frac {z^{k}}{\Gamma (k+1)}}=\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}=\exp(z).$

误差函数:

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

双曲余弦:

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

对应 $a=0,1,2$, ：

$\int _{0}^{z}E_{\alpha ,1}(-s^{2}){\mathrm {d} }s$ 有下列积分式

$\arctan(z)$,

${\tfrac {\sqrt {\pi }}{2}}\operatorname {erf} (z)$,

$\sin(z)$.

参考文献

• Mittag-Leffler, M.G.: Sur la nouvelle fonction E(x). C. R. Acad. Sci. Paris 137, 554–558 (1903)
• Mittag-Leffler, M.G.: Sopra la funzione E˛.x/. Rend. R. Acc. Lincei, (Ser. 5) 13, 3–5 (1904)
• Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V., Mittag-Leffler Functions, Related Topics and Applications (Springer, New York, 2014) （页面存档备份，存于互联网档案馆） 443 pages ISBN 978-3-662-43929-6

• Olver, F. W. J.; Maximon, L. C., 米塔-列夫勒函数, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (编), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248 
• Igor Podlubny. 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. 1998. ISBN 0-12-558840-2. 
• Kai Diethelm. 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. 2010. ISBN 978-3-642-14573-5.