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

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)
• [http://www.springer.com/gp/book/9783662439296 Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V., Mittag-Leffler Functions, Related Topics and Applications (Springer, New York, 2014)] 443 pages ISBN 978-3-662-43929-6

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