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1: 36.6 Scaling Relations

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$\text{<span class="ltx\_text hilite">cuspoids</span>: }\mathbf{y}(k)=(x_1{k}^{{\gamma }_{1K}},x_2{k}^{{\gamma }_{2K}},\mathrm{\dots },x_K{k}^{{\gamma }_{KK}}),$

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$\text{<span class="ltx\_text hilite">cuspoids</span>: }{\beta }_K={\displaystyle \frac{K}{2(K+2)}},$

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$\text{<span class="ltx\_text hilite">cuspoids</span>: }{\gamma }_{mK}=1-{\displaystyle \frac{m}{K+2}},$

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$\text{<span class="ltx\_text hilite">cuspoids</span>: }{\gamma }_K={\displaystyle \sum _{m=1}^K}{\gamma }_{mK}={\displaystyle \frac{K(K+3)}{2(K+2)}},$

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Table 36.6.1: Special cases of scaling exponents for cuspoids.

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singularity	$K$	${\beta }_K$	${\gamma }_{1K}$	${\gamma }_{2K}$	${\gamma }_{3K}$	${\gamma }_K$
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2: 36.12 Uniform Approximation of Integrals

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§36.12(i) General Theory for Cuspoids

… ►In the cuspoid case (one integration variable) … ►Define a mapping $u(t;\mathbf{y})$ by relating $f(u;\mathbf{y})$ to the normal form (36.2.1) of ${\mathrm{\Phi }}_K(t;\mathbf{x})$ in the following way: …with the $K+1$ functions $A(\mathbf{y})$ and $\mathbf{x}(\mathbf{y})$ determined by correspondence of the $K+1$ critical points of $f$ and ${\mathrm{\Phi }}_K$. …where $t_j(\mathbf{x})$, $1\le j\le K+1$, are the critical points of ${\mathrm{\Phi }}_K$, that is, the solutions (real and complex) of (36.4.1). …

3: 36.4 Bifurcation Sets

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Critical Points for Cuspoids

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36.4.1 $$\frac{\partial }{\partial t}{\mathrm{\Phi }}_K(t_j(\mathbf{x});\mathbf{x})=0.$$

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Symbols:

${\mathrm{\Phi }}_K(t;\mathbf{x})$: cuspoid catastrophe of codimension $K$ , $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $t$: variable, $K$: codimension and $t_j(\mathbf{x})$: solutions

Referenced by:

§36.11, §36.12(i), §36.4(i), §36.5(i), §36.5(ii)

Permalink:

http://dlmf.nist.gov/36.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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Bifurcation (Catastrophe) Set for Cuspoids

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36.4.3 $$\frac{{\partial }^2}{{\partial t}^2}{\mathrm{\Phi }}_K(t;\mathbf{x})=0.$$

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Symbols:

${\mathrm{\Phi }}_K(t;\mathbf{x})$: cuspoid catastrophe of codimension $K$ , $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $t$: variable and $K$: codimension

Permalink:

http://dlmf.nist.gov/36.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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4: 36.1 Special Notation

… ►The main functions covered in this chapter are cuspoid catastrophes ${\mathrm{\Phi }}_K(t;\mathbf{x})$; umbilic catastrophes with codimension three ${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;\mathbf{x})$, ${\mathrm{\Phi }}^{(\mathrm{H})}(s,t;\mathbf{x})$; canonical integrals ${\mathrm{\Psi }}_K\left(\mathbf{x}\right)$, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(\mathbf{x}\right)$, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(\mathbf{x}\right)$; diffraction catastrophes ${\mathrm{\Psi }}_K(\mathbf{x};k)$, ${\mathrm{\Psi }}^{(\mathrm{E})}(\mathbf{x};k)$, ${\mathrm{\Psi }}^{(\mathrm{H})}(\mathbf{x};k)$ generated by the catastrophes. …

5: 36.11 Leading-Order Asymptotics

… ►and far from the bifurcation set, the cuspoid canonical integrals are approximated by ►

36.11.2 $${\mathrm{\Psi }}_K\left(\mathbf{x}\right)=\sqrt{2\pi }\sum _{j=1}^{j_{\max }(\mathbf{x})}\exp \left(\mathrm{i}\left({\mathrm{\Phi }}_K(t_j(\mathbf{x});\mathbf{x})+\frac{1}{4}\pi {(-1)}^{j+K+1}\right)\right){\left|\frac{{\partial }^2{\mathrm{\Phi }}_K(t_j(\mathbf{x});\mathbf{x})}{{\partial t}^2}\right|}^{-1/2}(1+o\left(1\right)).$$

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Symbols:

${\mathrm{\Psi }}_K\left(\mathbf{x}\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Phi }}_K(t;\mathbf{x})$: cuspoid catastrophe of codimension $K$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $t$: variable, $K$: codimension and $t_j(\mathbf{x})$: solutions

Permalink:

http://dlmf.nist.gov/36.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11 and Ch.36

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6: 36.5 Stokes Sets

… ►Stokes sets are surfaces (codimension one) in $\mathbf{x}$ space, across which ${\mathrm{\Psi }}_K(\mathbf{x};k)$ or ${\mathrm{\Psi }}^{(\mathrm{U})}(\mathbf{x};k)$ acquires an exponentially-small asymptotic contribution (in $k$), associated with a complex critical point of ${\mathrm{\Phi }}_K$ or ${\mathrm{\Phi }}^{(\mathrm{U})}$. … ►

$\mathrm{\Re }\left({\mathrm{\Phi }}_K(t_j(\mathbf{x});\mathbf{x})-{\mathrm{\Phi }}_K(t_{\mu }(\mathbf{x});\mathbf{x})\right)=0,$

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§36.5(ii) Cuspoids

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7: 36.10 Differential Equations

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36.10.1 $${\mathrm{\Phi }}_K^{\prime }(-\mathrm{i}\frac{\partial }{\partial x_1};\mathbf{x}){\mathrm{\Psi }}_K\left(\mathbf{x}\right)=0,$$

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Symbols:

${\mathrm{\Psi }}_K\left(\mathbf{x}\right)$: canonical integral function, ${\mathrm{\Phi }}_K(t;\mathbf{x})$: cuspoid catastrophe of codimension $K$ , $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $K$: codimension and $x_{\mathrm{i}}$: real parameter

Referenced by:

§36.10(i)

Permalink:

http://dlmf.nist.gov/36.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(i), §36.10 and Ch.36

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${\mathrm{\Phi }}_s^{(\mathrm{U})}(-\mathrm{i}{\displaystyle \frac{\partial }{\partial x}},-\mathrm{i}{\displaystyle \frac{\partial }{\partial y}};\mathbf{x}){\mathrm{\Psi }}^{(\mathrm{U})}\left(\mathbf{x}\right)=0,$

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${\mathrm{\Phi }}_t^{(\mathrm{U})}(-\mathrm{i}{\displaystyle \frac{\partial }{\partial x}},-\mathrm{i}{\displaystyle \frac{\partial }{\partial y}};\mathbf{x}){\mathrm{\Psi }}^{(\mathrm{U})}\left(\mathbf{x}\right)=0,$

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${\mathrm{\Phi }}_s^{(\mathrm{U})}(s,t;\mathbf{x})={\displaystyle \frac{\partial }{\partial s}}{\mathrm{\Phi }}^{(\mathrm{U})}(s,t;\mathbf{x}),$

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${\mathrm{\Phi }}_t^{(\mathrm{U})}(s,t;\mathbf{x})={\displaystyle \frac{\partial }{\partial t}}{\mathrm{\Phi }}^{(\mathrm{U})}(s,t;\mathbf{x}).$

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8: 36.2 Catastrophes and Canonical Integrals

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Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension $K$

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36.2.1 $${\mathrm{\Phi }}_K(t;\mathbf{x})=t^{K+2}+\sum _{m=1}^K{x}_m{t}^m.$$

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Defines:

${\mathrm{\Phi }}_K(t;\mathbf{x})$: cuspoid catastrophe of codimension $K$

Symbols:

$n$: integer, $t$: variable, $K$: codimension and $x_{\mathrm{i}}$: real parameter

Referenced by:

§36.10(i), §36.12(i), §36.5(ii), §36.7(ii), §36.8

Permalink:

http://dlmf.nist.gov/36.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

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36.2.4 $${\mathrm{\Psi }}_K\left(\mathbf{x}\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\mathrm{i}{\mathrm{\Phi }}_K(t;\mathbf{x})\right)dt.$$

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Defines:

${\mathrm{\Psi }}_K\left(\mathbf{x}\right)$: canonical integral function

Symbols:

${\mathrm{\Phi }}_K(t;\mathbf{x})$: cuspoid catastrophe of codimension $K$ , $dx$: differential, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $t$: variable and $K$: codimension

Referenced by:

§36.12(i), §36.2(i), §36.7(ii), §36.8

Permalink:

http://dlmf.nist.gov/36.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

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36.2.10 $${\mathrm{\Psi }}_K(\mathbf{x};k)=\sqrt{k}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\mathrm{i}k{\mathrm{\Phi }}_K(t;\mathbf{x})\right)dt,$$ $k>0$.

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Defines:

${\mathrm{\Psi }}_K(\mathbf{x};k)$: diffraction catastrophe

Symbols:

${\mathrm{\Phi }}_K(t;\mathbf{x})$: cuspoid catastrophe of codimension $K$ , $dx$: differential, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $k$: variable, $t$: variable and $K$: codimension

Referenced by:

§36.12(i)

Permalink:

http://dlmf.nist.gov/36.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

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9: 36.7 Zeros

… ►The zeros of these functions are curves in $\mathbf{x}=(x,y,z)$ space; see Nye (2007) for ${\mathrm{\Phi }}_3$ and Nye (2006) for ${\mathrm{\Phi }}^{(\mathrm{H})}$.

10: Bibliography K

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An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives. Computer Physics Comm. 132 (1-2), pp. 142–165.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

§36.15(iii).

Encodings:

BibTeX

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