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DLMF: §36.6 Scaling Relations ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles

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36 Integrals with Coalescing Saddles Properties 36.5 Stokes Sets 36.7 Zeros

§36.6 Scaling Relations

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Keywords:: Pearcey integral, diffraction catastrophes, elliptic umbilic canonical integral, for diffraction catastrophes, hyperbolic umbilic canonical integral, scaling laws, swallowtail canonical integral
Permalink:: http://dlmf.nist.gov/36.6
See also:: Annotations for Ch.36

Diffraction Catastrophe Scaling

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See also:: Annotations for §36.6 and Ch.36

36.6.1		$\displaystyle\Psi_{K}(\mathbf{x};k)$	$\displaystyle=k^{\beta_{K}}\Psi_{K}\left(\mathbf{y}(k)\right),$
36.6.1		$\displaystyle\Psi^{(\mathrm{U})}(\mathbf{x};k)$	$\displaystyle=k^{\beta^{(\mathrm{U})}}\Psi^{(\mathrm{U})}\left(\mathbf{y}^{(% \mathrm{U})}(k)\right),$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\Psi_{\NVar{K}}(\NVar{\mathbf{x}};k)$ : diffraction catastrophe, $\Psi^{(\mathrm{U})}(\NVar{\mathbf{x}};\NVar{k})$ : umbilic canonical integral function, $\Psi^{(\mathrm{U})}\left(\NVar{\mathbf{x}}\right)$ : umbilic canonical integral function, $k$ : variable and $K$ : codimension Permalink: http://dlmf.nist.gov/36.6.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.6, §36.6 and Ch.36

where

36.6.2		$\displaystyle\text{cuspoids: }\mathbf{y}(k)$	$\displaystyle=\left(x_{1}k^{\gamma_{1K}},x_{2}k^{\gamma_{2K}},\dots,x_{K}k^{% \gamma_{KK}}\right),$
36.6.2		$\displaystyle\text{umbilics: }\mathbf{y}^{(\mathrm{U})}(k)$	$\displaystyle=\left(xk^{2/3},yk^{2/3},zk^{1/3}\right).$
ⓘ Symbols: $y$ : real parameter, $z$ : real parameter, $k$ : variable, $K$ : codimension, $x_{i}$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.6.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.6, §36.6 and Ch.36

Indices for $k$ -Scaling of Magnitude of $\Psi_{K}$ or $\Psi^{(\mathrm{U})}$ (Singularity Index)

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See also:: Annotations for §36.6 and Ch.36

36.6.3		$\displaystyle\text{cuspoids: }\beta_{K}$	$\displaystyle=\dfrac{K}{2(K+2)},$
36.6.3		$\displaystyle\text{umbilics: }\beta^{(\mathrm{U})}$	$\displaystyle=\frac{1}{3}.$
ⓘ Symbols: $K$ : codimension Permalink: http://dlmf.nist.gov/36.6.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.6, §36.6 and Ch.36

Indices for $k$ -Scaling of Coordinates $x_{m}$

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See also:: Annotations for §36.6 and Ch.36

36.6.4		$\displaystyle\text{cuspoids: }\gamma_{mK}$	$\displaystyle=1-\dfrac{m}{K+2},$
		$\displaystyle\text{umbilics: }\gamma_{x}^{(\mathrm{U})}$	$\displaystyle=\tfrac{2}{3},$
		$\displaystyle\gamma_{y}^{(\mathrm{U})}$	$\displaystyle=\tfrac{2}{3},$
		$\displaystyle\gamma_{z}^{(\mathrm{U})}$	$\displaystyle=\tfrac{1}{3}.$
ⓘ Symbols: $y$ : real parameter, $z$ : real parameter, $n$ : integer, $K$ : codimension and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.6.E4 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §36.6, §36.6 and Ch.36

Indices for $k$ -Scaling of $\mathbf{x}$ Hypervolume

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See also:: Annotations for §36.6 and Ch.36

36.6.5		$\displaystyle\text{cuspoids: }\gamma_{K}$	$\displaystyle=\sum_{m=1}^{K}\gamma_{mK}=\dfrac{K(K+3)}{2(K+2)},$
36.6.5		$\displaystyle\text{umbilics: }\gamma^{(\mathrm{U})}$	$\displaystyle=\sum_{m=1}^{3}\gamma_{m}^{(\mathrm{U})}=\tfrac{5}{3}.$
ⓘ Symbols: $n$ : integer and $K$ : codimension Permalink: http://dlmf.nist.gov/36.6.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.6, §36.6 and Ch.36

Table 36.6.1: Special cases of scaling exponents for cuspoids.

singularity	$K$	$\beta_{K}$	$\gamma_{1K}$	$\gamma_{2K}$	$\gamma_{3K}$	$\gamma_{K}$
fold	1	$\frac{1}{6}$	$\frac{2}{3}$	$-$	$-$	$\frac{2}{3}$
cusp	2	$\frac{1}{4}$	$\frac{3}{4}$	$\frac{1}{2}$	$-$	$\frac{5}{4}$
swallowtail	3	$\frac{3}{10}$	$\frac{4}{5}$	$\frac{3}{5}$	$\frac{2}{5}$	$\frac{9}{5}$

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Symbols:: $K$ : codimension
Permalink:: http://dlmf.nist.gov/36.6.T1
See also:: Annotations for §36.6, §36.6 and Ch.36

For the results in this section and more extensive lists of exponents see Berry (1977) and Varčenko (1976).

36.5 Stokes Sets 36.7 Zeros

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