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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.

33E12Mittag-Leffler functions and generalizations
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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