# elliptic umbilic canonical integral

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## 1—10 of 14 matching pages

##### 1: 36.2 Catastrophes and Canonical Integrals
###### CanonicalIntegrals
$\Psi^{(\mathrm{E})}\left(\boldsymbol{{0}}\right)=\tfrac{1}{3}\sqrt{\pi}\Gamma% \left(\tfrac{1}{6}\right)=3.28868,$
36.2.26 $\Psi^{(\mathrm{E})}\left(-\tfrac{1}{2}x\mp\tfrac{\sqrt{3}}{2}y,\pm\tfrac{\sqrt% {3}}{2}x-\tfrac{1}{2}y,z\right)=\Psi^{(\mathrm{E})}\left(x,y,z\right),$
36.2.28 $\Psi^{(\mathrm{E})}\left(0,0,z\right)=\overline{\Psi^{(\mathrm{E})}\left(0,0,-% z\right)}\\ =2\pi\sqrt{\frac{\pi z}{27}}\exp\left(\frac{2}{27}iz^{3}\right)\*\left(J_{-1/6% }\left(\frac{2}{27}z^{3}\right)+iJ_{1/6}\left(\frac{2}{27}z^{3}\right)\right),$ $z\geq 0$,
##### 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 …
36.10.13 $6\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{\partial x\partial y}-2iz\frac{% \partial\Psi^{(\mathrm{E})}}{\partial y}+y\Psi^{(\mathrm{E})}=0,$
36.10.14 $3\left(\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{{\partial x}^{2}}-\frac{{% \partial}^{2}\Psi^{(\mathrm{E})}}{{\partial y}^{2}}\right)+2iz\frac{\partial% \Psi^{(\mathrm{E})}}{\partial x}-x\Psi^{(\mathrm{E})}=0.$
36.10.17 $i\frac{\partial\Psi^{(\mathrm{E})}}{\partial z}=\frac{{\partial}^{2}\Psi^{(% \mathrm{E})}}{{\partial x}^{2}}+\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{{% \partial y}^{2}},$
##### 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: 36.8 Convergent Series Expansions
###### §36.8 Convergent Series Expansions
36.8.4 $\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)=2\pi^{2}\left(\dfrac{2}{3}\right)^{% 2/3}\sum\limits_{n=0}^{\infty}\dfrac{\left(-i(2/3)^{2/3}z\right)^{n}}{n!}\Re% \left(f_{n}\left(\dfrac{x+iy}{12^{1/3}},\dfrac{x-iy}{12^{1/3}}\right)\right),$
##### 6: 36.9 Integral Identities
36.9.9 $\left|\Psi^{(\mathrm{E})}\left(x,y,z\right)\right|^{2}=\frac{8\pi^{2}}{3^{2/3}% }\int_{0}^{\infty}\int_{0}^{2\pi}\Re\left(\mathrm{Ai}\left(\frac{1}{3^{1/3}}% \left(x+iy+2zu\exp\left(i\theta\right)+3u^{2}\exp\left(-2i\theta\right)\right)% \right)\*\mathrm{Bi}\left(\frac{1}{3^{1/3}}\left(x-iy+2zu\exp\left(-i\theta% \right)+3u^{2}\exp\left(2i\theta\right)\right)\right)\right)u\mathrm{d}u% \mathrm{d}\theta.{}$
36.11.7 $\Psi^{(\mathrm{E})}\left(0,0,z\right)=\frac{\pi}{z}\left(i+\sqrt{3}\exp\left(% \frac{4}{27}iz^{3}\right)+o\left(1\right)\right),$ $z\to\pm\infty$,
##### 8: 36.7 Zeros
###### §36.7(iii) EllipticUmbilicCanonicalIntegral
The zeros are lines in $\mathbf{x}=(x,y,z)$ space where $\operatorname{ph}\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)$ is undetermined. …Near $z=z_{n}$, and for small $x$ and $y$, the modulus $|\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)|$ has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose $z$ and $x$ repeat distances are given by …
##### 9: Errata
• 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.