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Pearcey integral

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1: 36.2 Catastrophes and Canonical Integrals
Ψ 2 is the Pearcey integral (Pearcey (1946)):
36.2.14 Ψ 2 ( x ) = P ( x 2 , x 1 ) = - exp ( i ( t 4 + x 2 t 2 + x 1 t ) ) d t .
2: 36.7 Zeros
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 .
3: 36.3 Visualizations of Canonical Integrals
See accompanying text
See accompanying text
Figure 36.3.1: Modulus of Pearcey integral | Ψ 2 ( x , y ) | . Magnify 3D Help
See accompanying text See accompanying text
Figure 36.3.13: Phase of Pearcey integral ph Ψ 2 ( x , y ) . Magnify
4: 36.6 Scaling Relations
§36.6 Scaling Relations
5: 36.9 Integral Identities
§36.9 Integral Identities
6: 36.11 Leading-Order Asymptotics
§36.11 Leading-Order Asymptotics
7: 36.8 Convergent Series Expansions
§36.8 Convergent Series Expansions
8: 36.12 Uniform Approximation of Integrals
For example, the diffraction catastrophe Ψ 2 ( x , y ; k ) defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function Ψ 1 ( ξ ( x , y ; k ) ) when k is large, provided that x and y are not small. … For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).
9: 36.5 Stokes Sets
§36.5(ii) Cuspoids
§36.5(iv) Visualizations
10: 36.10 Differential Equations
K = 2 , cusp: …