Past exam of the perturbation methods and applied complex variables course of the University of Oxford / 2009 / A / 3 / iii

In the overlap region ${ϵ}^{1/3}\ll x\ll 1$, write $X=x/{ϵ}^{1/3}$. The supplied Airy function asymptotics giveThe outer prefactor $e^{-x^2/4+x/2}$ tends to one in this overlap. ThereforeThe recessive condition $c_2=0$ gives $\beta =0$ and $\alpha =1/a_0=3^{2/3}\Gamma (2/3)$. Hence the leading inner answer isAway from the turning point, the matched WKB approximation isA useful multiplicative composite expansion through the turning region is $e^{-x^2/4+x/2}\mathrm{Ai}(x/{ϵ}^{1/3})/\mathrm{Ai}(0)$: it is finite and normalized at zero and matches the displayed outer expression for fixed positive $x$. It is a small-$ϵ$ approximation across the turning point, not a claim of relative uniform accuracy out to arbitrarily large $x$.