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36 Integrals with Coalescing SaddlesNotation

§36.1 Special Notation

(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.)

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