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

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

## Asymptotics along Symmetry Lines

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