36.2.E1.xml

$\Phi_{K}\left(t;\mathbf{x}\right)=t^{K+2}+\sum_{m=1}^{K}x_{m}t^{m}.$

36.2.E2.xml

$\Phi^{(\mathrm{E})}\left(s,t;\mathbf{x}\right)=s^{3}-3st^{2}+z(s^{2}+t^{2})+yt% +xs,$

36.2.E3.xml

$\Phi^{(\mathrm{H})}\left(s,t;\mathbf{x}\right)=s^{3}+t^{3}+zst+yt+xs,$

36.2.E4.xml

$\Psi_{K}\left(\mathbf{x}\right)=\int_{-\infty}^{\infty}\exp\left(i\Phi_{K}% \left(t;\mathbf{x}\right)\right)\,\mathrm{d}t.$

36.2.E5.xml

$\Psi^{(\mathrm{U})}\left(\mathbf{x}\right)=\int_{-\infty}^{\infty}\int_{-% \infty}^{\infty}\exp\left(i\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)% \right)\,\mathrm{d}s\,\mathrm{d}t,$

36.2.E6.xml

$\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)=2\sqrt{\ifrac{\pi}{3}}\,\exp\left(i% \left(\tfrac{4}{27}z^{3}+\tfrac{1}{3}xz-\tfrac{1}{4}\pi\right)\right)\int_{% \infty\exp\left(-7\pi i/12\right)}^{\infty\exp\left(\pi i/12\right)}\exp\left(% i\left(u^{6}+2zu^{4}+(z^{2}+x)u^{2}+\frac{y^{2}}{12u^{2}}\right)\right)\,% \mathrm{d}u,$

36.2.E7a.xml

$\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)=\dfrac{4\pi}{3^{1/3}}\exp\left(i% \left(\tfrac{2}{27}z^{3}-\tfrac{1}{3}xz\right)\right)\left(\exp\left(-i\dfrac{% \pi}{6}\right)\mathrm{F}_{+}(\mathbf{x})+\exp\left(i\dfrac{\pi}{6}\right)% \mathrm{F}_{-}(\mathbf{x})\right),$

36.2.E7b.xml

$\mathrm{F}_{\pm}(\mathbf{x})=\int_{0}^{\infty}\cos\left(ry\exp\left(\pm i% \dfrac{\pi}{6}\right)\right)\exp\left(2ir^{2}z\exp\left(\pm i\dfrac{\pi}{3}% \right)\right)\operatorname{Ai}\left(3^{2/3}r^{2}+3^{-1/3}\exp\left(\mp i% \dfrac{\pi}{3}\right)\left(\tfrac{1}{3}z^{2}-x\right)\right)\,\mathrm{d}r.$

36.2.E8.xml

$\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)=4\sqrt{\ifrac{\pi}{6}}\,\exp\left(i% \left(\tfrac{1}{27}z^{3}+\tfrac{1}{6}z(y+x)+\tfrac{1}{4}\pi\right)\right)\*% \int_{\infty\exp\left(5\pi i/12\right)}^{\infty\exp\left(\pi i/12\right)}\exp% \left(i\left(2u^{6}+2zu^{4}+\left(\tfrac{1}{2}z^{2}+x+y\right)u^{2}-\frac{(y-x% )^{2}}{24u^{2}}\right)\right)\,\mathrm{d}u,$

36.2.E9.xml

$\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)=\frac{2\pi}{3^{1/3}}\int_{\infty% \exp\left(5\pi i/6\right)}^{\infty\exp\left(\pi i/6\right)}\exp\left(i(s^{3}+% xs)\right)\operatorname{Ai}\left(\frac{zs+y}{3^{1/3}}\right)\,\mathrm{d}s.$

36.2.E10.xml

$\Psi_{K}(\mathbf{x};k)=\sqrt{k}\int_{-\infty}^{\infty}\exp\left(ik\Phi_{K}% \left(t;\mathbf{x}\right)\right)\,\mathrm{d}t,$

36.2.E11.xml

$\Psi^{(\mathrm{U})}(\mathbf{x};k)=k\int_{-\infty}^{\infty}\int_{-\infty}^{% \infty}\exp\left(ik\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)\right)\,% \mathrm{d}s\,\mathrm{d}t,$

36.2.E12.xml

$\Psi_{0}=\sqrt{\pi}\exp\left(i\frac{\pi}{4}\right).$

36.2.E13.xml

$\Psi_{1}\left(x\right)=\frac{2\pi}{3^{1/3}}\operatorname{Ai}\left(\frac{x}{3^{% 1/3}}\right).$

36.2.E14.xml

$\Psi_{2}\left(\mathbf{x}\right)=P(x_{2},x_{1})=\int_{-\infty}^{\infty}\exp% \left(\mathrm{i}(t^{4}+x_{2}t^{2}+x_{1}t)\right)\,\mathrm{d}t.$

36.2.E15.xml

$\Psi_{K}\left(\boldsymbol{{0}}\right)=\frac{2}{K+2}\Gamma\left(\frac{1}{K+2}% \right)\*\begin{cases}\exp\left(i\dfrac{\pi}{2(K+2)}\right),&K\text{ even,}\\ \cos\left(\dfrac{\pi}{2(K+2)}\right),&K\text{ odd}.\end{cases}$

36.2.E16a.xml

$\Psi_{1}\left(\boldsymbol{{0}}\right)=1.54669,$

36.2.E16b.xml

$\Psi_{2}\left(\boldsymbol{{0}}\right)=1.67481+\mathrm{i}\,0.69373$

36.2.E16c.xml

$\Psi_{3}\left(\boldsymbol{{0}}\right)=1.74646,$

36.2.E16d.xml

$\Psi_{4}\left(\boldsymbol{{0}}\right)=1.79222+\mathrm{i}\,0.48022.$

36.2.E17a.xml

$\frac{{\partial}^{p}}{{\partial x_{1}}^{p}}\Psi_{K}\left(\boldsymbol{{0}}% \right)=\frac{2}{K+2}\Gamma\left(\frac{p+1}{K+2}\right)\cos\left(\frac{\pi}{2}% \left(\frac{p+1}{K+2}+p\right)\right),$