With real critical points (36.4.1) ordered so that

 36.11.1 $t_{1}(\mathbf{x}) ⓘ Symbols: $t_{j}(\mathbf{x})$: solutions Permalink: http://dlmf.nist.gov/36.11.E1 Encodings: TeX, pMML, png See also: Annotations for §36.11 and Ch.36

and far from the bifurcation set, the cuspoid canonical integrals are approximated by

 36.11.2 $\Psi_{K}\left(\mathbf{x}\right)=\sqrt{2\pi}\sum\limits_{j=1}^{j_{\max}(\mathbf% {x})}\exp\left(i\left(\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)+\tfrac% {1}{4}\pi(-1)^{j+K+1}\right)\right)\left|\frac{{\partial}^{2}\Phi_{K}\left(t_{% j}(\mathbf{x});\mathbf{x}\right)}{{\partial t}^{2}}\right|^{-1/2}(1+o\left(1% \right)).$

## Asymptotics along Symmetry Lines

 36.11.3 $\Psi_{2}\left(0,y\right)=\begin{cases}\sqrt{\ifrac{\pi}{y}}\left(\exp\left(% \tfrac{1}{4}i\pi\right)+o\left(1\right)\right),&y\to+\infty,\\ \sqrt{\ifrac{\pi}{|y|}}\exp\left(-\tfrac{1}{4}i\pi\right)\left(1+i\sqrt{2}\exp% \left(-\frac{1}{4}iy^{2}\right)+o\left(1\right)\right),&y\to-\infty.\end{cases}$
 36.11.4 $\displaystyle\Psi_{3}\left(x,0,0\right)$ $\displaystyle=\frac{\sqrt{2\pi}}{(5|x|^{3})^{1/8}}\begin{cases}\exp\left(-2% \sqrt{2}(\ifrac{x}{5})^{5/4}\right)\left(\cos\left(2\sqrt{2}(\ifrac{x}{5})^{5/% 4}-\tfrac{1}{8}\pi\right)+o\left(1\right)\right),&x\to+\infty,\\ \cos\left(4(\ifrac{|x|}{5})^{5/4}-\tfrac{1}{4}\pi\right)+o\left(1\right),&x\to% -\infty.\end{cases}$ 36.11.5 $\displaystyle\Psi_{3}\left(0,y,0\right)$ $\displaystyle=\overline{\Psi_{3}(0,-y,0)}=\exp\left(\tfrac{1}{4}i\pi\right)% \sqrt{\ifrac{\pi}{y}}\left(1-(i/\sqrt{3})\exp\left(\tfrac{3}{2}i(\ifrac{2y}{5}% )^{5/3}\right)+o\left(1\right)\right),$ $y\to+\infty$. 36.11.6 $\displaystyle\Psi_{3}\left(0,0,z\right)$ $\displaystyle=\frac{\Gamma\left(\tfrac{1}{3}\right)}{|z|^{1/3}\sqrt{3}}+\begin% {cases}o\left(1\right),&z\to+\infty,\\ \dfrac{2\sqrt{\pi}5^{1/4}}{(3|z|)^{3/4}}\left(\cos\left(\dfrac{2}{3}\left(% \dfrac{3|z|}{5}\right)^{5/2}-\dfrac{1}{4}\pi\right)+o\left(1\right)\right),&z% \to-\infty.\end{cases}$
 36.11.7 $\displaystyle\Psi^{(\mathrm{E})}\left(0,0,z\right)$ $\displaystyle=\frac{\pi}{z}\left(i+\sqrt{3}\exp\left(\frac{4}{27}iz^{3}\right)% +o\left(1\right)\right),$ $z\to\pm\infty$, 36.11.8 $\displaystyle\Psi^{(\mathrm{H})}\left(0,0,z\right)$ $\displaystyle=\frac{2\pi}{z}\left(1-\frac{i}{\sqrt{3}}\exp\left(\frac{1}{27}iz% ^{3}\right)+o\left(1\right)\right),$ $z\to\pm\infty$.