The Mittag-Leffler function $E_{\alpha \beta}$ is a complex function which depends on two complex parameters $\alpha$ and $\beta$ It may be defined by the following series when the real part of $\alpha$ is strictly positive: $$E_{\alpha \beta} (z) = \sum_{k=0}^\infty {z^k \over \Gamma (\alpha k + \beta)}$$ In this case, the series converges for all values of the argument $z$ so the Mittag-Leffler function is an entire function.