Asymptotic Expansion of Mellin Transforms in the Complex Plane

In an earlier paper by the author [A. Sidi, SIAM J. Math. Anal., 16 (1985), pp. 896–906], asymptotic expansions for Mellin transforms f̂(z) = ∫∞ 0 t z−1f(t) dt as z → ∞, with z real and positive, were derived. In particular, it was shown there that, for certain classes of functions uk(t), k = 0, 1, . . . , that form asymptotic scales as t → ∞, if f(t) ∼ ∑∞ k=0 Akuk(t) as t → ∞, then f̂(z) ∼ ∑∞ k=0 Akûk(z) as z → ∞. In this note, we show that, for two of the cases considered there, f̂(z) ∼ ∑∞ k=0 Akûk(z) as z → ∞, also when z is complex and |Iz| ≤ η(Rz)c, for some c ∈ (0, 1) and some fixed, but otherwise arbitrary, η > 0. AMS Subject Classification: 30E10, 30E15, 33C10, 41A60, 44A15