canonical denominator (or numerator)

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2: 36.3 Visualizations of Canonical Integrals
§36.3(ii) Canonical Integrals: Phase
In Figure 36.3.13(a) points of confluence of phase contours are zeros of $\Psi_{2}\left(x,y\right)$; similarly for other contour plots in this subsection. In Figure 36.3.13(b) points of confluence of all colors are zeros of $\Psi_{2}\left(x,y\right)$; similarly for other density plots in this subsection. …
3: 1.12 Continued Fractions
$A_{n}$ and $B_{n}$ are called the $n$th (canonical) numerator and denominator respectively. …
1.12.8 $C_{n}-C_{n-1}=\frac{(-1)^{n-1}\prod^{n}_{k=1}a_{k}}{B_{n-1}B_{n}},$ $n=1,2,3,\dots$,
5: 36.10 Differential Equations
§36.10(i) Equations for $\Psi_{K}\left(\mathbf{x}\right)$
In terms of the normal form (36.2.1) the $\Psi_{K}\left(\mathbf{x}\right)$ satisfy the operator equation … $K=3$, swallowtail: … 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 …
6: 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 $\left|\Psi^{(\mathrm{E})}\left(x,y,z\right)\right|^{2}=\frac{8\pi^{2}}{3^{2/3}% }\int_{0}^{\infty}\int_{0}^{2\pi}\Re\left(\operatorname{Ai}\left(\frac{1}{3^{1% /3}}\left(x+iy+2zu\exp\left(i\theta\right)+3u^{2}\exp\left(-2i\theta\right)% \right)\right)\*\operatorname{Bi}\left(\frac{1}{3^{1/3}}\left(x-iy+2zu\exp% \left(-i\theta\right)+3u^{2}\exp\left(2i\theta\right)\right)\right)\right)u\,% \mathrm{d}u\,\mathrm{d}\theta.$
7: 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. …
8: 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)
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 $\Psi^{(\mathrm{H})}\left(0,0,z\right)=\frac{2\pi}{z}\left(1-\frac{i}{\sqrt{3}}% \exp\left(\frac{1}{27}iz^{3}\right)+o\left(1\right)\right),$ $z\to\pm\infty$.
$\Psi_{K}\left(\mathbf{x}\right)=\dfrac{2}{K+2}\sum\limits_{n=0}^{\infty}\exp% \left(i\dfrac{\pi(2n+1)}{2(K+2)}\right)\Gamma\left(\dfrac{2n+1}{K+2}\right)a_{% 2n}(\mathbf{x}),$ $K$ 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 $\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)=2\pi^{2}\left(\dfrac{2}{3}\right)^{% 2/3}\sum\limits_{n=0}^{\infty}\dfrac{\left(-i(2/3)^{2/3}z\right)^{n}}{n!}\Re% \left(f_{n}\left(\dfrac{x+iy}{12^{1/3}},\dfrac{x-iy}{12^{1/3}}\right)\right),$