§36.1 Special Notation

Keywords:: canonical integrals, diffraction catastrophes
Permalink:: http://dlmf.nist.gov/36.1

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

$l,m,n$	integers.
$k,t,s$	real or complex variables.
$K$	codimension.
$\mathbf{x}$	$\{ x_{1},x_{2},\dots,x_{K}\}$ , where $x_{1},x_{2},\dots,x_{K}$ are real parameters; also $x_{1}=x$ , $x_{2}=y$ , $x_{3}=z$ when $K\leq 3$ .
$\mathop{\mathrm{Ai}\/}\nolimits$ , $\mathop{\mathrm{Bi}\/}\nolimits$	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.)