36 Integrals with Coalescing Saddles Notation 36 Integrals with Coalescing Saddles 36.2 Catastrophes and Canonical Integrals

§36.1 Special Notation

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Keywords:: canonical integrals, diffraction catastrophes, notation
Referenced by:: Erratum (V1.0.20) for Section 36.1 Special Notation
Permalink:: http://dlmf.nist.gov/36.1
See also:: Annotations for Ch.36

(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 $K\leq 3$ .
$\operatorname{Ai}$ , $\operatorname{Bi}$	Airy functions (§9.2).

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.)

36 Integrals with Coalescing Saddles 36.2 Catastrophes and Canonical Integrals

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