Mittag-Leffler

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

• {{ #if: |{{{2}}}|H.J. Haubold}}{{#if: A.M. Mathai|{{#if: R.K. Saxena|, {{ #if: |{{{2}}}|A.M. Mathai}}{{#if: |, {{ #if: |{{{2}}}|R.K. Saxena}}{{#if: |, [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]{{#if: |, [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]] and [[Mathematician:{{{author6}}}|{{ #if: |{{{2}}}|{{{author6}}}}}]]| and [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]]}}| and [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]}}| and {{ #if: |{{{2}}}|R.K. Saxena}}}}| and {{ #if: |{{{2}}}|A.M. Mathai}}}}|}}: [[Paper:H.J. Haubold/Mittag-Leffler Functions and Their Applications{{#if: |/Volume {{{volume}}}|}}{{#if: |/{{{edpage}}}}}|Mittag-Leffler Functions and Their Applications{{#if: |: Volume {{{volume}}}|}}{{#if: |: {{{eddisplay}}} (2011)| ({{#if: |{{{ed}}} ed., }}2011)}}]]{{#if: |, {{{publisher}}}|}}{{#if: |, ISBN {{{isbn}}}|}}{{#if: | ... [[{{{prev}}}|(previous)]]|}}{{#if: E (0,1)(z)=1/(1-z) for abs(z) less than 1 | ... (next)|}}{{#if: |: Entry: {{#if: |[[{{{entryref}}}|{{{entry}}}]]|{{{entry}}}}}|}}: $(1.1)$ (has $\beta=1$ and uses the notation $E_{\alpha}$) and $(1.2)$