# canonical integrals

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##### 1: 36.2 Catastrophes and Canonical Integrals
###### CanonicalIntegrals
36.2.24 $\Psi^{(\mathrm{U})}\left(x,y,z\right)=\overline{\Psi^{(\mathrm{U})}\left(x,y,-% z\right)},$ $\mathrm{U=E,H}$.
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 …
36.10.13 $6\frac{\,{\partial}^{2}\Psi^{(\mathrm{E})}}{\,\partial x\,\partial y}-2iz\frac% {\partial\Psi^{(\mathrm{E})}}{\partial y}+y\Psi^{(\mathrm{E})}=0,$
36.10.15 $3\frac{{\partial}^{2}\Psi^{(\mathrm{H})}}{{\partial x}^{2}}+iz\frac{\partial% \Psi^{(\mathrm{H})}}{\partial y}-x\Psi^{(\mathrm{H})}=0,$
36.10.16 $3\frac{{\partial}^{2}\Psi^{(\mathrm{H})}}{{\partial y}^{2}}+iz\frac{\partial% \Psi^{(\mathrm{H})}}{\partial x}-y\Psi^{(\mathrm{H})}=0.$
36.10.17 $i\frac{\partial\Psi^{(\mathrm{E})}}{\partial z}=\frac{{\partial}^{2}\Psi^{(% \mathrm{E})}}{{\partial x}^{2}}+\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{{% \partial y}^{2}},$
##### 4: 36.1 Special Notation
###### §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
36.9.1 $|\Psi_{1}\left(x\right)|^{2}=2^{5/3}\int_{0}^{\infty}\Psi_{1}\left(2^{2/3}(3u^% {2}+x)\right)\,\mathrm{d}u;$
36.9.8 $\left|\Psi^{(\mathrm{H})}\left(x,y,z\right)\right|^{2}=8\pi^{2}\left(\frac{2}{% 9}\right)^{1/3}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\operatorname{Ai}% \left(\left(\frac{4}{3}\right)^{1/3}(x+zv+3u^{2})\right)\operatorname{Ai}\left% (\left(\frac{4}{3}\right)^{1/3}(y+zu+3v^{2})\right)\,\mathrm{d}u\,\mathrm{d}v.$
36.9.9 $\left|\Psi^{(\mathrm{E})}\left(x,y,z\right)\right|^{2}=\frac{8\pi^{2}}{3^{2/3}% }\int_{0}^{\infty}\int_{0}^{2\pi}\Re\left(\operatorname{Ai}\left(\frac{1}{3^{1% /3}}\left(x+iy+2zu\exp\left(i\theta\right)+3u^{2}\exp\left(-2i\theta\right)% \right)\right)\*\operatorname{Bi}\left(\frac{1}{3^{1/3}}\left(x-iy+2zu\exp% \left(-i\theta\right)+3u^{2}\exp\left(2i\theta\right)\right)\right)\right)u\,% \mathrm{d}u\,\mathrm{d}\theta.$
##### 6: 36.6 Scaling Relations
###### §36.6 Scaling Relations
$\Psi_{K}(\mathbf{x};k)=k^{\beta_{K}}\Psi_{K}\left(\mathbf{y}(k)\right),$
$\Psi^{(\mathrm{U})}(\mathbf{x};k)=k^{\beta^{(\mathrm{U})}}\Psi^{(\mathrm{U})}% \left(\mathbf{y}^{(\mathrm{U})}(k)\right),$
###### Indices for $k$-Scaling of Magnitude of $\Psi_{K}$ or $\Psi^{(\mathrm{U})}$ (Singularity Index)
and far from the bifurcation set, the cuspoid canonical integrals are approximated by …
36.11.5 $\Psi_{3}\left(0,y,0\right)=\overline{\Psi_{3}(0,-y,0)}=\exp\left(\tfrac{1}{4}i% \pi\right)\sqrt{\ifrac{\pi}{y}}\left(1-(i/{\sqrt{3}})\exp\left(\tfrac{3}{2}i(% \ifrac{2y}{5})^{5/3}\right)+o\left(1\right)\right),$ $y\to+\infty$.
36.11.7 $\Psi^{(\mathrm{E})}\left(0,0,z\right)=\frac{\pi}{z}\left(i+\sqrt{3}\exp\left(% \frac{4}{27}iz^{3}\right)+o\left(1\right)\right),$ $z\to\pm\infty$,
36.11.8 $\Psi^{(\mathrm{H})}\left(0,0,z\right)=\frac{2\pi}{z}\left(1-\frac{i}{\sqrt{3}}% \exp\left(\frac{1}{27}iz^{3}\right)+o\left(1\right)\right),$ $z\to\pm\infty$.
##### 8: 36.8 Convergent Series Expansions
###### §36.8 Convergent Series Expansions
$\Psi_{K}\left(\mathbf{x}\right)=\dfrac{2}{K+2}\sum\limits_{n=0}^{\infty}\exp% \left(i\dfrac{\pi(2n+1)}{2(K+2)}\right)\Gamma\left(\dfrac{2n+1}{K+2}\right)a_{% 2n}(\mathbf{x}),$ $K$ even,
For multinomial power series for $\Psi_{K}\left(\mathbf{x}\right)$, see Connor and Curtis (1982).
36.8.3 $\dfrac{3^{2/3}}{4\pi^{2}}\Psi^{(\mathrm{H})}\left(3^{1/3}\mathbf{x}\right)=% \operatorname{Ai}\left(x\right)\operatorname{Ai}\left(y\right)\sum\limits_{n=0% }^{\infty}(-3^{-1/3}iz)^{n}\dfrac{c_{n}(x)c_{n}(y)}{n!}+\operatorname{Ai}\left% (x\right)\operatorname{Ai}'\left(y\right)\sum\limits_{n=2}^{\infty}(-3^{-1/3}% iz)^{n}\dfrac{c_{n}(x)d_{n}(y)}{n!}+\operatorname{Ai}'\left(x\right)% \operatorname{Ai}\left(y\right)\sum\limits_{n=2}^{\infty}(-3^{-1/3}iz)^{n}% \dfrac{d_{n}(x)c_{n}(y)}{n!}+\operatorname{Ai}'\left(x\right)\operatorname{Ai}% '\left(y\right)\sum\limits_{n=1}^{\infty}(-3^{-1/3}iz)^{n}\dfrac{d_{n}(x)d_{n}% (y)}{n!},$
36.8.4 $\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)=2\pi^{2}\left(\dfrac{2}{3}\right)^{% 2/3}\sum\limits_{n=0}^{\infty}\dfrac{\left(-i(2/3)^{2/3}z\right)^{n}}{n!}\Re% \left(f_{n}\left(\dfrac{x+iy}{12^{1/3}},\dfrac{x-iy}{12^{1/3}}\right)\right),$
##### 10: 36.7 Zeros
###### §36.7(ii) Cusp CanonicalIntegral
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 …