new variable $x$, we have $$\frac{d^2W}{dx^2}=\left(\lambda^2 g(x) - \frac{1}{4x^2}\right)W \;,$$ where $g(x)=\frac{(x-x_1)(x-x_2)}{4x^2}$. Using the ideas explained in @imj, we shall be - eventually - interested in the asymptotics for $z=\mu_N+\sigma_N s$, or $x=z/\lambda=x_2+\sigma_N s/\lambda$ of $F_N(z)$. Let us now define an auxiliary variable $\upsilon$ by $$\begin{aligned}
\int_{\beta}^{\upsilon}(\tau^2-\beta^2)^{1/2}d\tau &= \int_{x_2}^x g^{1/2}(t)dt \hspace{1cm} \text{if } x_2 \leq x
< \infty \;,
\\
\int_{-\beta}^{\upsilon}(\beta^2-\tau^2)^{1/2}d\tau &= \int_{x_1}^x (-g)^{1/2}(t)dt \hspace{1cm} \text{if } x_1
\leq x \leq x_2 \;.\end{aligned}$$

We limit $x$ to this range because of the technically important following point: ${\sigma_N}/\lambda$ tends to zero faster than $x_2-x_1$ does, and so, when $s$ is bounded below, $x$ will stay in the range $(x_1,\infty)$ for all $N$ greater than a certain $N_0$. This is shown in **A2**, along with the closely related fact that we can focus on $\upsilon\geq 0$. Our analysis is based on section 3 of @olver80, where he builds on @olver75, in which he expands Whittaker functions in terms of parabolic cylinder functions. The condition ${\upsilon}\geq 0$ is critical, since Olver’s expansions depend on the sign of $\upsilon$. Therefore, **A2** entitles us to focus on only one specific form of these. From (3.10) p.219 in @olver80, one has $$\begin{aligned}
{W_{\kappa,\lambda}}(\lambda x)=
(2\lambda)^{1/4}&\{\lambda(2+\beta^2/2)/e\}^{\lambda(1+\beta^2/4)}
&\times\left(\frac{{\upsilon}^2-\beta^2}{x^2-4lx+4}\right)^{1/4}x^{1/2}
