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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.
𝐱 {x1,x2,,xK}, where x1,x2,,xK are real parameters; also x1=x, x2=y, x3=z when K3.
Ai, Bi Airy functions (§9.2).

The main functions covered in this chapter are cuspoid catastrophes ΦK(t;𝐱); umbilic catastrophes with codimension three Φ(E)(s,t;𝐱), Φ(H)(s,t;𝐱); canonical integrals ΨK(𝐱), Ψ(E)(𝐱), Ψ(H)(𝐱); diffraction catastrophes ΨK(𝐱;k), Ψ(E)(𝐱;k), Ψ(H)(𝐱;k) generated by the catastrophes. (There is no standard nomenclature for these functions.)