# §36.6 Scaling Relations

## Diffraction Catastrophe Scaling

 36.6.1 $\displaystyle\Psi_{K}(\mathbf{x};k)$ $\displaystyle=k^{\beta_{K}}\Psi_{K}\left(\mathbf{y}(k)\right),$ $\displaystyle\Psi^{(\mathrm{U})}(\mathbf{x};k)$ $\displaystyle=k^{\beta^{(\mathrm{U})}}\Psi^{(\mathrm{U})}\left(\mathbf{y}^{(% \mathrm{U})}(k)\right),$

where

 36.6.2 $\displaystyle\text{cuspoids: }\mathbf{y}(k)$ $\displaystyle=\left(x_{1}k^{\gamma_{1K}},x_{2}k^{\gamma_{2K}},\dots,x_{K}k^{% \gamma_{KK}}\right),$ $\displaystyle\text{umbilics: }\mathbf{y}^{(\mathrm{U})}(k)$ $\displaystyle=\left(xk^{2/3},yk^{2/3},zk^{1/3}\right).$

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

 36.6.3 $\displaystyle\text{cuspoids: }\beta_{K}$ $\displaystyle=\dfrac{K}{2(K+2)},$ $\displaystyle\text{umbilics: }\beta^{(\mathrm{U})}$ $\displaystyle=\frac{1}{3}.$ ⓘ 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 $x_{m}$

 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

 36.6.5 $\displaystyle\text{cuspoids: }\gamma_{K}$ $\displaystyle=\sum_{m=1}^{K}\gamma_{mK}=\dfrac{K(K+3)}{2(K+2)},$ $\displaystyle\text{umbilics: }\gamma^{(\mathrm{U})}$ $\displaystyle=\sum_{m=1}^{3}\gamma_{m}^{(\mathrm{U})}=\tfrac{5}{3}.$ ⓘ 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

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