# §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 $\mathop{\Phi_{K}\/}\nolimits\!\left(t;\mathbf{x}\right)$; umbilic catastrophes with codimension three $\mathop{\Phi^{(\mathrm{E})}\/}\nolimits\!\left(s,t;\mathbf{x}\right)$, $\mathop{\Phi^{(\mathrm{H})}\/}\nolimits\!\left(s,t;\mathbf{x}\right)$; canonical integrals $\mathop{\Psi_{K}\/}\nolimits\!\left(\mathbf{x}\right)$, $\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!\left(\mathbf{x}\right)$, $\mathop{\Psi^{(\mathrm{H})}\/}\nolimits\!\left(\mathbf{x}\right)$; diffraction catastrophes $\mathop{\Psi_{K}\/}\nolimits\!(\mathbf{x};k)$, $\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!(\mathbf{x};k)$, $\mathop{\Psi^{(\mathrm{H})}\/}\nolimits\!(\mathbf{x};k)$ generated by the catastrophes. (There is no standard nomenclature for these functions.)