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1: 36.2 Catastrophes and Canonical Integrals
Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension K
Special cases: K = 1 , fold catastrophe; K = 2 , cusp catastrophe; K = 3 , swallowtail catastrophe.
Normal Forms for Umbilic Catastrophes with Codimension K = 3
Canonical Integrals
Diffraction Catastrophes
2: 36.4 Bifurcation Sets
Bifurcation (Catastrophe) Set for Cuspoids
Bifurcation (Catastrophe) Set for Umbilics
K = 1 , fold bifurcation set: …
See accompanying text
Figure 36.4.1: Bifurcation set of cusp catastrophe. Magnify
See accompanying text
Figure 36.4.2: Bifurcation set of swallowtail catastrophe. Magnify
3: 36.7 Zeros
§36.7(i) Fold 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 .
The zeros of these functions are curves in 𝐱 = ( x , y , z ) space; see Nye (2007) for Φ 3 and Nye (2006) for Φ ( H ) .
4: 36.6 Scaling Relations
§36.6 Scaling Relations
Diffraction Catastrophe Scaling
Ψ K ( 𝐱 ; k ) = k β K Ψ K ( 𝐲 ( k ) ) ,
Ψ ( U ) ( 𝐱 ; k ) = k β ( U ) Ψ ( U ) ( 𝐲 ( U ) ( k ) ) ,
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
fold 1 1 6 2 3 2 3
5: 36.10 Differential Equations
K = 1 , fold: (36.10.1) becomes Airy’s equation (§9.2(i)) K = 1 , fold: (36.10.6) is an identity. …
Φ s ( U ) ( s , t ; 𝐱 ) = s Φ ( U ) ( s , t ; 𝐱 ) ,
Φ t ( U ) ( s , t ; 𝐱 ) = t Φ ( U ) ( s , t ; 𝐱 ) .
6: 36.5 Stokes Sets
Stokes sets are surfaces (codimension one) in 𝐱 space, across which Ψ K ( 𝐱 ; k ) or Ψ ( U ) ( 𝐱 ; k ) acquires an exponentially-small asymptotic contribution (in k ), associated with a complex critical point of Φ K or Φ ( U ) . …
See accompanying text
Figure 36.5.2: Swallowtail catastrophe with z < 0 . Magnify
See accompanying text
Figure 36.5.3: Swallowtail catastrophe with z = 0 . Magnify
7: 21.8 Abelian Functions
An Abelian function is a 2 g -fold periodic, meromorphic function of g complex variables. …
8: 36.1 Special Notation
§36.1 Special Notation
The main functions covered in this chapter are cuspoid catastrophes Φ K ( t ; 𝐱 ) ; umbilic catastrophes with codimension three Φ ( E ) ( s , t ; 𝐱 ) , Φ ( H ) ( s , t ; 𝐱 ) ; canonical integrals Ψ K ( 𝐱 ) , Ψ ( E ) ( 𝐱 ) , Ψ ( H ) ( 𝐱 ) ; diffraction catastrophes Ψ K ( 𝐱 ; k ) , Ψ ( E ) ( 𝐱 ; k ) , Ψ ( H ) ( 𝐱 ; k ) generated by the catastrophes. …
9: 20 Theta Functions
Chapter 20 Theta Functions
10: 36.9 Integral Identities
§36.9 Integral Identities