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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)= n=0 z n /Γ(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:
33E12Mittag-Leffler functions and generalizations
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)