查看“︁米塔-列夫勒函数”︁的源代码


[[File:Mittag-Leffler function 2D plot.gif|thumb|Mittag-Leffler function 2D plot]]
[[File:Mittag-Leffler function Imaginary 3D plot1.gif|thumb|Mittag-Leffler function Imaginary 3D plot1]]
[[File:Mittag-Leffler function Imaginary 3D plot2.gif|thumb|Mittag-Leffler function Imaginary 3D plot2]]
'''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

<math>E_{a,b}(z)=\sum_{k=0}^{\infty}\frac{z^k}{\Gamma(ak+b)}</math>

==特例==
对应 <math>a=0,1/2,1,2</math> 有


:<math>E_{0,1}(z) = \sum_{k=0}^\infty z^k = \frac{1}{1-z}.</math>

[[指数函数]]:
:<math>E_{1,1}(z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma (k + 1)} = \sum_{k=0}^\infty \frac{z^k}{k!} = \exp(z).</math>

[[误差函数]]:
:<math>E_{1/2,1}(z) = \exp(z^2)\operatorname{erfc}(-z).</math>

[[双曲余弦]]:
:<math>E_{2,1}(z) = \cosh(\sqrt{z}).</math>

对应 <math>a=0,1,2</math>, ： 
:<math>\int_0^{z}E_{\alpha,1}(-s^2){\mathrm d}s</math> 有下列积分式


:<math>\arctan(z)</math>, 

:<math>\tfrac{\sqrt{\pi}}{2}\operatorname{erf}(z)</math>,

:<math>\sin(z)</math>.




==参考文献==
<references/>
* 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)] {{Wayback|url=http://www.springer.com/gp/book/9783662439296 |date=20180326112951 }} 443 pages ISBN 978-3-662-43929-6

*{{dlmf|first=F. W. J. |last=Olver|authorlink1=Frank W. J. Olver|first2=L. C. |last2=Maximon|id=10.46}}
* {{cite book|title=Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications|series=Mathematics in Science and Engineering|author=Igor Podlubny|publisher=Academic Press|year=1998|isbn=0-12-558840-2|chapter=chapter 1}}
* {{cite book|author=Kai Diethelm|title=The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type|location=Heidelberg and New York|publisher=Springer-Verlag|year=2010|series=Lecture notes in mathematics|isbn=978-3-642-14573-5|chapter=chapter 4}}

[[Category:特殊函数]]