36 Integrals with Coalescing Saddles Properties 36.5 Stokes Sets 36.7 Zeros

§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, : variable and : 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:: : real parameter, : real parameter, : variable, : codimension, $x_{i}$ : real parameter and : real parameter
Permalink:: http://dlmf.nist.gov/36.6.E2
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¶ Indices for -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:: : codimension
Permalink:: http://dlmf.nist.gov/36.6.E3
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¶ Indices for -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:: : real parameter, : real parameter, : integer, : codimension and : real parameter
Permalink:: http://dlmf.nist.gov/36.6.E4
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¶ Indices for -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:: : integer and : 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		$\beta_{K}$	$\gamma_{{1K}}$	$\gamma_{{2K}}$	$\gamma_{{3K}}$	$\gamma_{K}$
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	$\frac{3}{10}$	$\frac{4}{5}$	$\frac{3}{5}$	$\frac{2}{5}$	$\frac{9}{5}$