Difference between revisions of "Mittag-Leffler"
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(Created page with "The Mittag-Leffler function $E_{\alpha, \beta}$ is defined by the series $$E_{\alpha, \beta}(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{\Gamma(\alpha k + \beta)},$$ where...") |
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| − | The Mittag-Leffler function $E_{\alpha, \beta}$ is defined by the series | + | The Mittag-Leffler function $E_{\alpha, \beta}$ is defined for $z, \alpha, \beta \in \mathbb{C}$ with $\mathrm{Re}(\alpha), \mathrm{Re}(\beta) > 0$ by the series |
$$E_{\alpha, \beta}(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{\Gamma(\alpha k + \beta)},$$ | $$E_{\alpha, \beta}(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{\Gamma(\alpha k + \beta)},$$ | ||
where $\Gamma$ denotes the [[gamma]] function. | where $\Gamma$ denotes the [[gamma]] function. | ||
=Properties= | =Properties= | ||
| + | [[E (0,1)(z)=1/(1-z) for abs(z) less than 1]]<br /> | ||
| + | [[E(1,1)(z)=exp(z)]]<br /> | ||
| + | [[E(2,1)(z)=cosh(sqrt(z))]]<br /> | ||
| + | [[E(2,1)(-z^2)=cos(z)]]<br /> | ||
=References= | =References= | ||
| + | * {{PaperReference|Mittag-Leffler Functions and Their Applications|2011|H.J. Haubold|author2=A.M. Mathai|author3=R.K. Saxena|next=E (0,1)(z)=1/(1-z) for abs(z) less than 1}}: $(1.1)$ (has $\beta=1$ and uses the notation $E_{\alpha}$) and $(1.2)$ | ||
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Latest revision as of 21:35, 2 January 2018
The Mittag-Leffler function $E_{\alpha, \beta}$ is defined for $z, \alpha, \beta \in \mathbb{C}$ with $\mathrm{Re}(\alpha), \mathrm{Re}(\beta) > 0$ by the series $$E_{\alpha, \beta}(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{\Gamma(\alpha k + \beta)},$$ where $\Gamma$ denotes the gamma function.
Properties
E (0,1)(z)=1/(1-z) for abs(z) less than 1
E(1,1)(z)=exp(z)
E(2,1)(z)=cosh(sqrt(z))
E(2,1)(-z^2)=cos(z)
References
- H.J. Haubold, A.M. Mathai and R.K. Saxena: Mittag-Leffler Functions and Their Applications (2011)... (next): $(1.1)$ (has $\beta=1$ and uses the notation $E_{\alpha}$) and $(1.2)$
