Difference between revisions of "Mittag-Leffler"

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=Properties=
 
=Properties=
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[[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 />
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[[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=findme}}: $(1.1)$ (has $\beta=1$ and uses the notation $E_{\alpha}$) and $(1.2)$
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* {{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