§36.6 Scaling Relations

Keywords:: Pearcey integral, diffraction catastrophes, elliptic umbilic canonical integral, hyperbolic umbilic canonical integral, scaling laws, swallowtail canonical integral
Permalink:: http://dlmf.nist.gov/36.6

¶ Diffraction Catastrophe Scaling

36.6.1

$\mathop{\Psi _{{K}}\/}\nolimits\!(\mathbf{x};k)=k^{{\beta _{K}}}\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{y}(k)\right),$

$\mathop{\Psi^{{(\mathrm{U})}}\/}\nolimits\!(\mathbf{x};k)=k^{{\beta^{{(\mathrm{U})}}}}\mathop{\Psi^{{(\mathrm{U})}}\/}\nolimits\!\left(\mathbf{y}^{{(\mathrm{U})}}(k)\right),$

Symbols:: $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\Psi^{{(\mathrm{U})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical umbilic integral, $\mathop{\Psi _{{K}}\/}\nolimits\!(\mathbf{x};k)$ : diffraction catastrophe, $\mathop{\Psi^{{(\mathrm{U})}}\/}\nolimits\!(\mathbf{x};k)$ : umbilic canonical integral function, $k$ : variable and $K$ : codimension
Permalink:: http://dlmf.nist.gov/36.6.E1
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where

36.6.2

$\text{cuspoids: }\mathbf{y}(k)=\left(x_{1}k^{{\gamma _{{1K}}}},x_{2}k^{{\gamma _{{2K}}}},\dots,x_{K}k^{{\gamma _{{KK}}}}\right),$

$\text{umbilics: }\mathbf{y}^{{(\mathrm{U})}}(k)=\left(xk^{{2/3}},yk^{{2/3}},zk^{{1/3}}\right).$

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
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¶ Indices for $k$ -Scaling of Magnitude of $\mathop{\Psi _{{K}}\/}\nolimits$ or $\mathop{\Psi^{{(\mathrm{U})}}\/}\nolimits$ (Singularity Index)

36.6.3

$\text{cuspoids: }\beta _{K}=\dfrac{K}{2(K+2)},$

$\text{umbilics: }\beta^{{(\mathrm{U})}}=\frac{1}{3}.$

Symbols:: $K$ : codimension
Permalink:: http://dlmf.nist.gov/36.6.E3
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¶ Indices for $k$ -Scaling of Coordinates $x_{m}$

36.6.4

$\text{cuspoids: }\gamma _{{mK}}=1-\dfrac{m}{K+2},$

$\text{umbilics: }\gamma _{x}^{{(\mathrm{U})}}=\tfrac{2}{3},$

$\gamma _{y}^{{(\mathrm{U})}}=\tfrac{2}{3},$

$\gamma _{z}^{{(\mathrm{U})}}=\tfrac{1}{3}.$

Symbols:: $y$ : real parameter, $z$ : real parameter, $n$ : integer, $K$ : codimension and $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.6.E4
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¶ Indices for $k$ -Scaling of $\mathbf{x}$ Hypervolume

36.6.5

$\text{cuspoids: }\gamma _{K}=\sum\limits _{{m=1}}^{K}\gamma _{{mK}}=\dfrac{K(K+3)}{2(K+2)},$

$\text{umbilics: }\gamma^{{(\mathrm{U})}}=\sum\limits _{{m=1}}^{3}\gamma _{m}^{{(\mathrm{U})}}=\tfrac{5}{3}.$

Symbols:: $n$ : integer and $K$ : codimension
Permalink:: http://dlmf.nist.gov/36.6.E5
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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	$\frac{3}{10}$	$\frac{4}{5}$	$\frac{3}{5}$	$\frac{2}{5}$	$\frac{9}{5}$