complex case. We then turn to the problem of the asymptotic behavior of $c_{\phi}$, and the technical points we have to verify for @sosh results to hold.

The following remarks outline the differences between the analysis we present here and the one done in @imj.

Remarks on adaptation of the original proof
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### Complex case {#errorcontrolproblem}

To show that (\[whattoshow1\]) and (\[whattoshow2\]) held true, @imj essentially reduced his problem to studying the solution of a “perturbed" Airy equation and used tools from @olver to carefully study it. One point that was used repeatedly was that the turning points of the equation were bounded away from one another when $n,N$ were large. This is not true anymore in the case we consider, and we show how to get around this difficulty. So we do not work with a perturbed Airy equation anymore, but rather with Whittaker functions, which have a close relationship to Laguerre polynomials, and their expansion in terms of parabolic cylinder functions (see **A9** for some background information on special functions). In @olver80, the case we are interested in was studied in detail, giving us most of the tools we need to show (\[whattoshow2\]). Using @olver75, we reinterpret 