# §36.1 Special Notation

(For other notation see Notation for the Special Functions.)

$l,m,n$ integers. real or complex variables. codimension. $\{x_{1},x_{2},\dots,x_{K}\}$, where $x_{1},x_{2},\dots,x_{K}$ are real parameters; also $x_{1}=x$, $x_{2}=y$, $x_{3}=z$ when $K\leq 3$. Airy functions (§9.2). complex conjugate.

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. (There is no standard nomenclature for these functions.)