Airy contour and Fresnel normalization

[Entirety and the ODE] This is [F1]; in particular the Taylor coefficients obey the Airy recurrence and $A$ has no singularity on $\mathbb{R}$.

[Positive asymptotics] For $Z>0$ shift the vertical contour: $t=\sqrt{Z}+iy$; the horizontal joining terms vanish by the bound $\exp (-x{R}^2+O(x^3+Zx))$, so $A(Z)=\frac{1}{2\pi }{e}^{-2Z^{3\mathrm{/}2}\mathrm{/}3}{\int }_{\mathbb{R}}{e}^{-\sqrt{Z}{y}^2-i{y}^3\mathrm{/}3}dy$. Substituting $y=Z^{-1\mathrm{/}4}u$ and using that the sine part is odd with $\mathrm{\mid }\cos (u^3\mathrm{/}(3Z^{3\mathrm{/}4}))-1\mathrm{\mid }\le u^6\mathrm{/}(18Z^{3\mathrm{/}2})$ together with the finite Gaussian moments [F3] gives the display of (c); for $A^{\mathrm{\prime }}$ differentiate the entire contour representation and insert the extra amplitude $-t=-(\sqrt{Z}+i{Z}^{-1\mathrm{/}4}u)$ on the shifted contour. The term linear in $u$ has zero Gaussian integral; its first nonzero contribution after expansion of the cubic phase, and the quadratic phase remainder, are both of relative order $Z^{-3\mathrm{/}2}$ by finite Gaussian moments. This gives the second formula.

[Negative asymptotics] For $z=-Z$ put $t=\sqrt{Z}s$ in [F2], with $\lambda =Z^{3\mathrm{/}2}$ and $\phi (s)=s^3\mathrm{/}3-s$. Its stationary points are $s=\pm 1$, with phase values $\mp 2\mathrm{/}3$ and second derivatives $\pm 2$. Choose smooth compact cutoffs there and exact local coordinates $v$ with $\phi =\phi (s_0)+\sigma v^2\mathrm{/}2$, $\sigma =\mathrm{sign}{s}_0$. The transformed amplitude $b$ is smooth compactly supported, and $b(0)=1\mathrm{/}\sqrt{2}$ for amplitude $1$. The Fresnel identity applied to its Fourier inversion gives $\int e^{i\lambda \sigma v^2\mathrm{/}2}b(v)dv=e^{i\sigma \pi \mathrm{/}4}\sqrt{2\pi \mathrm{/}\lambda }[b(0)+O({\lambda }^{-1}{\int }_{\mathbb{R}}{\xi }^2\mathrm{\mid }\widehat{b}(\xi )\mathrm{\mid }d\xi )],$ where a fixed Fourier convention only changes the constant. Indeed expand the exact Fourier multiplier $e^{-i\sigma {\xi }^2\mathrm{/}(2\lambda )}$ using $\mathrm{\mid }{e}^{i\alpha }-1\mathrm{\mid }\le \mathrm{\mid }\alpha \mathrm{\mid }$; the displayed moment is finite because $b$ is smooth with compact support. The Fresnel identity can first be damped by $e^{-\delta v^2}$ and then passed to the oscillatory limit. Outside the two patches integrate twice by parts with $(i\lambda {\phi }^{\mathrm{\prime }}{)}^{-1}d\mathrm{/}ds$; the resulting amplitudes are integrable since ${\phi }^{\mathrm{\prime }}\sim s^2$ at infinity, including for amplitude $s$, and the finite boundaries vanish or are absorbed by the cutoffs. These terms are $O({\lambda }^{-2})$. Summing the two stationary contributions and restoring the factor $\sqrt{Z}\mathrm{/}(2\pi )$ gives the sine expression in (d) with absolute error $O(Z^{-7\mathrm{/}4})$. For $A^{\mathrm{\prime }}$ differentiate once in [F2] and use amplitude $is$, restoring the factor $Z\mathrm{/}(2\pi )$; the same estimate gives the cosine expression and absolute error $O(Z^{-5\mathrm{/}4})$. No higher ordinary oscillatory derivative is used. This proves (b) and (d), while steps 1.1 and 2.1 prove (a) and (c).