# cuspoids

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##### 1: 36.6 Scaling Relations
$\text{cuspoids: }\mathbf{y}(k)=\left(x_{1}k^{\gamma_{1K}},x_{2}k^{\gamma_{2K}}% ,\dots,x_{K}k^{\gamma_{KK}}\right),$
$\text{cuspoids: }\beta_{K}=\dfrac{K}{2(K+2)},$
$\text{cuspoids: }\gamma_{mK}=1-\dfrac{m}{K+2},$
$\text{cuspoids: }\gamma_{K}=\sum\limits_{m=1}^{K}\gamma_{mK}=\dfrac{K(K+3)}{2(% K+2)},$
##### 2: 36.12 Uniform Approximation of Integrals
###### §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 $\Phi_{K}\left(t;\mathbf{x}\right)$ 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 $\Phi_{K}$. …where $t_{j}(\mathbf{x})$, $1\leq j\leq K+1$, are the critical points of $\Phi_{K}$, that is, the solutions (real and complex) of (36.4.1). …
##### 3: 36.4 Bifurcation Sets
###### Critical Points for Cuspoids
36.4.1 $\frac{\partial}{\partial t}\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)=0.$
###### Bifurcation (Catastrophe) Set for Cuspoids
36.4.3 $\frac{{\partial}^{2}}{{\partial t}^{2}}\Phi_{K}\left(t;\mathbf{x}\right)=0.$
##### 4: 36.1 Special Notation
The main functions covered in this chapter are cuspoid catastrophes $\Phi_{K}\left(t;\mathbf{x}\right)$; umbilic catastrophes with codimension three $\Phi^{(\mathrm{E})}\left(s,t;\mathbf{x}\right)$, $\Phi^{(\mathrm{H})}\left(s,t;\mathbf{x}\right)$; canonical integrals $\Psi_{K}\left(\mathbf{x}\right)$, $\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)$, $\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)$; diffraction catastrophes $\Psi_{K}(\mathbf{x};k)$, $\Psi^{(\mathrm{E})}(\mathbf{x};k)$, $\Psi^{(\mathrm{H})}(\mathbf{x};k)$ generated by the catastrophes. …
and far from the bifurcation set, the cuspoid canonical integrals are approximated by
36.11.2 $\Psi_{K}\left(\mathbf{x}\right)=\sqrt{2\pi}\sum\limits_{j=1}^{j_{\max}(\mathbf% {x})}\exp\left(i\left(\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)+\tfrac% {1}{4}\pi(-1)^{j+K+1}\right)\right)\left|\frac{{\partial}^{2}\Phi_{K}\left(t_{% j}(\mathbf{x});\mathbf{x}\right)}{{\partial t}^{2}}\right|^{-1/2}(1+o\left(1% \right)).$
##### 6: 36.5 Stokes Sets
Stokes sets are surfaces (codimension one) in $\mathbf{x}$ space, across which $\Psi_{K}(\mathbf{x};k)$ or $\Psi^{(\mathrm{U})}(\mathbf{x};k)$ acquires an exponentially-small asymptotic contribution (in $k$), associated with a complex critical point of $\Phi_{K}$ or $\Phi^{(\mathrm{U})}$. …
$\Re\left(\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)-\Phi_{K}\left(t_{% \mu}(\mathbf{x});\mathbf{x}\right)\right)=0,$
##### 7: 36.10 Differential Equations
36.10.1 $\Phi_{K}'\left(-i\frac{\partial}{\partial x_{1}};\mathbf{x}\right)\Psi_{K}% \left(\mathbf{x}\right)=0,$
${\Phi_{s}}^{(\mathrm{U})}\left(-i\frac{\partial}{\partial x},-i\frac{\partial}% {\partial y};\mathbf{x}\right)\Psi^{(\mathrm{U})}\left(\mathbf{x}\right)=0,$
${\Phi_{t}}^{(\mathrm{U})}\left(-i\frac{\partial}{\partial x},-i\frac{\partial}% {\partial y};\mathbf{x}\right)\Psi^{(\mathrm{U})}\left(\mathbf{x}\right)=0,$
${\Phi_{s}}^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)=\frac{\partial}{\partial s% }\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right),$
${\Phi_{t}}^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)=\frac{\partial}{\partial t% }\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right).$
##### 8: 36.2 Catastrophes and Canonical Integrals
###### Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension $K$
36.2.1 $\Phi_{K}\left(t;\mathbf{x}\right)=t^{K+2}+\sum_{m=1}^{K}x_{m}t^{m}.$
##### 9: 36.7 Zeros
The zeros of these functions are curves in $\mathbf{x}=(x,y,z)$ space; see Nye (2007) for $\Phi_{3}$ and Nye (2006) for $\Phi^{(\mathrm{H})}$.
##### 10: Bibliography K
• N. P. Kirk, J. N. L. Connor, and C. A. Hobbs (2000) 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.