\{U(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda})+{\epsilon}_1(\lambda^2,\beta^2,{\upsilon})\} \; ,\end{aligned}$$ where, if ${\mathbf{E}}$ and ${\mathbf{M}}$ are the weight and modulus functions associated with $U$ in @olver75 (p.156), we have, according to @olver80 (3.11) p.219, $$\label{error}
{\epsilon}_1(\lambda^2,\beta^2,{\upsilon})={\mathbf{E}}^{-1}(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda})
{\mathbf{M}}(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda})\;\mathrm{O}(\lambda^{-2/3})$$ **uniformly** with respect to $\beta\in [0,B]$ and ${\upsilon}\in [0,\infty)$, $B$ being an arbitrary positive constant. We recall that the main relationship between $U$, ${\mathbf{E}}$ and ${\mathbf{M}}$ : for $b \leq 0$ and $x\geq 0$, $|U(b,x)|\leq {\mathbf{E}}^{-1}(b,x){\mathbf{M}}(b,x)$.

We now show that we have uniform boundedness on $[s_0,\infty)$. The pointwise convergence result will be a straightforward consequence of the arguments we need to develop to solve this first problem.

Uniform Boundedness
-------------------

Following up on the previous displays, if $n,N$ are large enough so that ${\upsilon}\geq 0$, we have $$\begin{gathered}
\label{Wbound}
\left|{W_{\kappa,\lambda}}(\lambda x)\right|\leq (2\lambda)^{1/4}\{\lambda(2+\beta^2/2)/e\}^{\lambda(1+\beta^2/4)}
\times\left(\frac{{\upsilon}^2-\beta^2}{x^2-4lx+4}\right)^{1/4}x^{1/2}{\mathbf{M}}{\mathbf{E}^{-1}}(1+\mathrm{O}(\lambda^{-2/3})) \;,\end{gathered}$$ where we omitted the argument $(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda})$ for readability purposes. Our plan is now to transform this upper bound into a somewhat similar one, involving the modulus and weight function associated with the Airy function, which have the advantage of having only one parameter and known asymptotics.

To carry out this program, we need to split the investigation into two parts: first $s\geq 0$ or ${\upsilon}\geq \beta$. This will allow us to find an $s_1 \geq 0$ 