The author considers the asymptotic behavior of the Mittag-Leffler function
for large complex and fixed real positive . An asymptotic expansion in inverse powers of 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 , showing the appearance of exponentially small terms in its asymptotic expansion.
The author analyzes the Stokes phenomena for varying arg (and fixed ) showing the appearance of an error function in the optimally truncated remainder. Two regions for are considered in this analysis: and . Finally, the author also analyzes the Stokes phenomena for varying and fixed arg, described again by an error function.