I need estimates for the parabolic cylinder functions $U(a,z)$ (first studied by Whittaker).
Most work in the literature focuses on $a$ real. As it happens, I am interested in $U(a,z)$ on a strip in the complex plane:
$|Re(a)|<=1/2$, $|Re(z)|$ bounded, $|Im(a)|$ large ($|Im(a)|\geq max(200,10 |Im(z)|)$, say).
What is known then? What can be proven easily by a non-specialist using known techniques? What I need is a relatively simple estimate for $U(a,z)$ in this range; I do not necessarily need a very good error term (an order-of-magnitude estimate is fine), but I absolutely need all constants in the bound for the error term to be explicit.
(A 1961 paper by Olver gives bounds that are qualitatively strong enough throughout the range, but, unfortunately, the constants are not explicit, and, since the bounds are based on asymptotic series (in Poincaré's sense), making the constants explicit is non-trivial. A 1965 paper, also by Olver, seems useful, but the range in which it works out things in detail is essentially disjoint from the above.)
