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36 Integrals with Coalescing SaddlesProperties

§36.6 Scaling Relations

Diffraction Catastrophe Scaling

36.6.1 ΨK(x;k) =kβKΨK(y(k)),
Ψ(U)(x;k) =kβ(U)Ψ(U)(y(U)(k)),

where

36.6.2 cuspoids: y(k) =(x1kγ1K,x2kγ2K,,xKkγKK),
umbilics: y(U)(k) =(xk2/3,yk2/3,zk1/3).

Indices for k-Scaling of Magnitude of ΨK or Ψ(U) (Singularity Index)

36.6.3 cuspoids: βK =K2(K+2),
umbilics: β(U) =13.

Indices for k-Scaling of Coordinates xm

36.6.4 cuspoids: γmK =1-mK+2,
umbilics: γx(U) =23,
γy(U) =23,
γz(U) =13.

Indices for k-Scaling of x Hypervolume

36.6.5 cuspoids: γK =m=1KγmK
=K(K+3)2(K+2),
umbilics: γ(U) =m=13γm(U)
=53.
Table 36.6.1: Special cases of scaling exponents for cuspoids.
singularity K βK γ1K γ2K γ3K γK
fold 1 16 23 - - 23
cusp 2 14 34 12 - 54
swallowtail 3 310 45 35 25 95

For the results in this section and more extensive lists of exponents see Berry (1977) and Varčenko (1976).