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Exponential asymptotics of the Mittag-Leffler function. (English) Zbl 1049.33018
The author considers the asymptotic behavior of the Mittag-Leffler function $E_a(z)=\sum_{n=0}^\infty z^n/\Gamma(an+1)$ for large complex $$z$$ and fixed real positive $$a$$. An asymptotic expansion in inverse powers of $$z$$ is obtained from a recurrence and an integral representation for the error term is given. From this integral the author determines the optimal truncation point and the exponentially improved asymptotics of $$E_a(z)$$, showing the appearance of exponentially small terms in its asymptotic expansion.
The author analyzes the Stokes phenomena for varying arg$$(z)$$ (and fixed $$a$$) showing the appearance of an error function in the optimally truncated remainder. Two regions for $$a$$ are considered in this analysis: $$0<a<1$$ and $$a>1$$. Finally, the author also analyzes the Stokes phenomena for varying $$a$$ and fixed arg$$(z)$$, described again by an error function.

##### MSC:
 33E12 Mittag-Leffler functions and generalizations 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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