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1: 36.2 Catastrophes and Canonical Integrals
Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension K
Canonical Integrals
2: 36.7 Zeros
§36.7(ii) Cusp Canonical Integral
Table 36.7.1: Zeros of cusp diffraction catastrophe to 5D.
Zeros { x y } inside, and zeros [ x y ] outside, the cusp x 2 = 8 27 | y | 3 .
Away from the z -axis and approaching the cusp lines (ribs) (36.4.11), the lattice becomes distorted and the rings are deformed, eventually joining to form “hairpins” whose arms become the pairs of zeros (36.7.1) of the cusp canonical integral. …
3: 36.4 Bifurcation Sets
K = 1 , fold bifurcation set: … K = 2 , cusp bifurcation set: … Swallowtail cusp lines (ribs): … Elliptic umbilic cusp lines (ribs): … Hyperbolic umbilic cusp line (rib): …
4: 36.10 Differential Equations
§36.10 Differential Equations
K = 2 , cusp: … K = 2 , cusp: … K = 3 , swallowtail: … In terms of the normal forms (36.2.2) and (36.2.3), the Ψ ( U ) ( x ) satisfy the following operator equations …
5: 36.5 Stokes Sets
§36.5(ii) Cuspoids
K = 2 . Cusp
The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set: …
§36.5(iii) Umbilics
§36.5(iv) Visualizations
6: Bibliography C
  • B. C. Carlson (1965) On computing elliptic integrals and functions. J. Math. and Phys. 44, pp. 36–51.
  • J. N. L. Connor, P. R. Curtis, and D. Farrelly (1983) A differential equation method for the numerical evaluation of the Airy, Pearcey and swallowtail canonical integrals and their derivatives. Molecular Phys. 48 (6), pp. 1305–1330.
  • J. N. L. Connor and D. Farrelly (1981) Molecular collisions and cusp catastrophes: Three methods for the calculation of Pearcey’s integral and its derivatives. Chem. Phys. Lett. 81 (2), pp. 306–310.
  • J. N. L. Connor (1973) Evaluation of multidimensional canonical integrals in semiclassical collision theory. Molecular Phys. 26 (6), pp. 1371–1377.
  • D. Cvijović and J. Klinowski (1999) Integrals involving complete elliptic integrals. J. Comput. Appl. Math. 106 (1), pp. 169–175.
  • 7: 36.6 Scaling Relations
    §36.6 Scaling Relations
    Ψ K ( x ; k ) = k β K Ψ K ( y ( k ) ) ,
    Ψ ( U ) ( x ; k ) = k β ( U ) Ψ ( U ) ( y ( U ) ( k ) ) ,
    Indices for k -Scaling of Magnitude of Ψ K or Ψ ( U ) (Singularity Index)
    Table 36.6.1: Special cases of scaling exponents for cuspoids.
    singularity K β K γ 1 K γ 2 K γ 3 K γ K
    cusp 2 1 4 3 4 1 2 - 5 4
    8: Bibliography B
  • G. E. Barr (1968) A note on integrals involving parabolic cylinder functions. SIAM J. Appl. Math. 16 (1), pp. 71–74.
  • M. V. Berry and C. J. Howls (1994) Overlapping Stokes smoothings: Survival of the error function and canonical catastrophe integrals. Proc. Roy. Soc. London Ser. A 444, pp. 201–216.
  • M. V. Berry (1975) Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces. J. Phys. A 8 (4), pp. 566–584.
  • W. G. C. Boyd (1973) The asymptotic analysis of canonical problems in high-frequency scattering theory. II. The circular and parabolic cylinders. Proc. Cambridge Philos. Soc. 74, pp. 313–332.
  • P. J. Bushell (1987) On a generalization of Barton’s integral and related integrals of complete elliptic integrals. Math. Proc. Cambridge Philos. Soc. 101 (1), pp. 1–5.