# hyperbolic umbilic canonical integral

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## 1—10 of 14 matching pages

##### 1: 36.2 Catastrophes and Canonical Integrals
###### CanonicalIntegrals
$\Psi^{(\mathrm{H})}\left(\boldsymbol{{0}}\right)=\tfrac{1}{3}{\Gamma}^{2}\left% (\tfrac{1}{3}\right)=2.39224.$
36.2.21 $\Psi^{(\mathrm{H})}\left(x,y,0\right)=\frac{4\pi^{2}}{3^{2/3}}\operatorname{Ai% }\left(\frac{x}{3^{1/3}}\right)\operatorname{Ai}\left(\frac{y}{3^{1/3}}\right).$
36.2.29 $\Psi^{(\mathrm{H})}\left(0,0,z\right)=\overline{\Psi^{(\mathrm{H})}\left(0,0,-% z\right)}=\frac{2^{1/3}}{\sqrt{3}}\exp\left(\frac{1}{27}iz^{3}\right)\Psi^{(% \mathrm{E})}\left(0,0,-\frac{z}{2^{2/3}}\right),$ $-\infty.
##### 3: 36.10 Differential Equations
In terms of the normal forms (36.2.2) and (36.2.3), the $\Psi^{(\mathrm{U})}\left(\mathbf{x}\right)$ satisfy the following operator equations …
36.10.14 $3\left(\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{{\partial x}^{2}}-\frac{{% \partial}^{2}\Psi^{(\mathrm{E})}}{{\partial y}^{2}}\right)+2iz\frac{\partial% \Psi^{(\mathrm{E})}}{\partial x}-x\Psi^{(\mathrm{E})}=0.$
36.10.15 $3\frac{{\partial}^{2}\Psi^{(\mathrm{H})}}{{\partial x}^{2}}+iz\frac{\partial% \Psi^{(\mathrm{H})}}{\partial y}-x\Psi^{(\mathrm{H})}=0,$
36.10.16 $3\frac{{\partial}^{2}\Psi^{(\mathrm{H})}}{{\partial y}^{2}}+iz\frac{\partial% \Psi^{(\mathrm{H})}}{\partial x}-y\Psi^{(\mathrm{H})}=0.$
36.10.18 $i\frac{\partial\Psi^{(\mathrm{H})}}{\partial z}=\frac{\,{\partial}^{2}\Psi^{(% \mathrm{H})}}{\,\partial x\,\partial y}.$
##### 4: 36.1 Special Notation
The main functions covered in this chapter are cuspoid catastrophes $\Phi_{K}\left(t;\mathbf{x}\right)$; umbilic catastrophes with codimension three $\Phi^{(\mathrm{E})}\left(s,t;\mathbf{x}\right)$, $\Phi^{(\mathrm{H})}\left(s,t;\mathbf{x}\right)$; canonical integrals $\Psi_{K}\left(\mathbf{x}\right)$, $\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)$, $\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)$; diffraction catastrophes $\Psi_{K}(\mathbf{x};k)$, $\Psi^{(\mathrm{E})}(\mathbf{x};k)$, $\Psi^{(\mathrm{H})}(\mathbf{x};k)$ generated by the catastrophes. …
##### 5: 36.9 Integral Identities
###### §36.9 Integral Identities
36.9.8 $\left|\Psi^{(\mathrm{H})}\left(x,y,z\right)\right|^{2}=8\pi^{2}\left(\frac{2}{% 9}\right)^{1/3}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\operatorname{Ai}% \left(\left(\frac{4}{3}\right)^{1/3}(x+zv+3u^{2})\right)\operatorname{Ai}\left% (\left(\frac{4}{3}\right)^{1/3}(y+zu+3v^{2})\right)\,\mathrm{d}u\,\mathrm{d}v.$
##### 6: 36.11 Leading-Order Asymptotics
###### §36.11 Leading-Order Asymptotics
36.11.8 $\Psi^{(\mathrm{H})}\left(0,0,z\right)=\frac{2\pi}{z}\left(1-\frac{i}{\sqrt{3}}% \exp\left(\frac{1}{27}iz^{3}\right)+o\left(1\right)\right),$ $z\to\pm\infty$.
##### 7: 36.8 Convergent Series Expansions
###### §36.8 Convergent Series Expansions
36.8.3 $\dfrac{3^{2/3}}{4\pi^{2}}\Psi^{(\mathrm{H})}\left(3^{1/3}\mathbf{x}\right)=% \operatorname{Ai}\left(x\right)\operatorname{Ai}\left(y\right)\sum\limits_{n=0% }^{\infty}(-3^{-1/3}iz)^{n}\dfrac{c_{n}(x)c_{n}(y)}{n!}+\operatorname{Ai}\left% (x\right)\operatorname{Ai}'\left(y\right)\sum\limits_{n=2}^{\infty}(-3^{-1/3}% iz)^{n}\dfrac{c_{n}(x)d_{n}(y)}{n!}+\operatorname{Ai}'\left(x\right)% \operatorname{Ai}\left(y\right)\sum\limits_{n=2}^{\infty}(-3^{-1/3}iz)^{n}% \dfrac{d_{n}(x)c_{n}(y)}{n!}+\operatorname{Ai}'\left(x\right)\operatorname{Ai}% '\left(y\right)\sum\limits_{n=1}^{\infty}(-3^{-1/3}iz)^{n}\dfrac{d_{n}(x)d_{n}% (y)}{n!},$
##### 8: Errata
• Figures 36.3.9, 36.3.10, 36.3.11, 36.3.12

Scales were corrected in all figures. The interval $-8.4\leq\frac{x-y}{\sqrt{2}}\leq 8.4$ was replaced by $-12.0\leq\frac{x-y}{\sqrt{2}}\leq 12.0$ and $-12.7\leq\frac{x+y}{\sqrt{2}}\leq 4.2$ replaced by $-18.0\leq\frac{x+y}{\sqrt{2}}\leq 6.0$. All plots and interactive visualizations were regenerated to improve image quality.

Reported 2016-09-12 by Dan Piponi.

• Figures 36.3.18, 36.3.19, 36.3.20, 36.3.21

The scaling error reported on 2016-09-12 by Dan Piponi also applied to contour and density plots for the phase of the hyperbolic umbilic canonical integrals. Scales were corrected in all figures. The interval $-8.4\leq\frac{x-y}{\sqrt{2}}\leq 8.4$ was replaced by $-12.0\leq\frac{x-y}{\sqrt{2}}\leq 12.0$ and $-12.7\leq\frac{x+y}{\sqrt{2}}\leq 4.2$ replaced by $-18.0\leq\frac{x+y}{\sqrt{2}}\leq 6.0$. All plots and interactive visualizations were regenerated to improve image quality.

Reported 2016-09-28.

• Equation (36.10.14)
36.10.14 $3\left(\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{{\partial x}^{2}}-\frac{{% \partial}^{2}\Psi^{(\mathrm{E})}}{{\partial y}^{2}}\right)+2\mathrm{i}z\frac{% \partial\Psi^{(\mathrm{E})}}{\partial x}-x\Psi^{(\mathrm{E})}=0$

Originally this equation appeared with $\frac{\partial\Psi^{(\mathrm{H})}}{\partial x}$ in the second term, rather than $\frac{\partial\Psi^{(\mathrm{E})}}{\partial x}$.

Reported 2010-04-02.

##### 10: 36.7 Zeros
###### §36.7(iv) Swallowtail and HyperbolicUmbilicCanonicalIntegrals
The zeros of these functions are curves in $\mathbf{x}=(x,y,z)$ space; see Nye (2007) for $\Phi_{3}$ and Nye (2006) for $\Phi^{(\mathrm{H})}$.