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Zbl 1049.33018
Paris, R.B.
Exponential asymptotics of the Mittag-Leffler function. (English)
[J] Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 458, No. 2028, 3041-3052 (2002). ISSN 1364-5021; ISSN 1471-2946/e

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. \par 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.
[José L. Lopez (Pamplona)]

MSC 2000:: *33E12 Mittag-Leffler functions and generalizations
41A60 Asymptotic problems in approximation

Keywords: Mittag-Leffler function; Stokes's phenomenon; exponentially improved asymptotics

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