Exponential asymptotics of the Mittag–Leffler function

We consider the exponentially improved asymptotics of the Mittag–Leffler function E⊂a(z) and show how certain exponentially small terms make their appearance in the asymptotic expansion of E⊂a(z) when |z| is large. A new terminant function is introduced (a generalized Mittag–Leffler function E⊂a,b(z)) and the Stokes phenomenon in both θ = arg z and the parameter a is examined.