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1: 36.6 Scaling Relations
cuspoids y ( k ) = ( x 1 k γ 1 K , x 2 k γ 2 K , , x K k γ K K ) ,
cuspoids β K = K 2 ( K + 2 ) ,
cuspoids γ m K = 1 - m K + 2 ,
cuspoids γ K = m = 1 K γ m K = K ( K + 3 ) 2 ( K + 2 ) ,
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
2: 36.12 Uniform Approximation of Integrals
§36.12(i) General Theory for Cuspoids
In the cuspoid case (one integration variable) … Define a mapping u ( t ; y ) by relating f ( u ; y ) to the normal form (36.2.1) of Φ K ( t ; x ) in the following way: …with the K + 1 functions A ( y ) and x ( y ) determined by correspondence of the K + 1 critical points of f and Φ K . …where t j ( x ) , 1 j K + 1 , are the critical points of Φ K , that is, the solutions (real and complex) of (36.4.1). …
3: 36.4 Bifurcation Sets
Critical Points for Cuspoids
Bifurcation (Catastrophe) Set for Cuspoids
4: 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. …
5: 36.11 Leading-Order Asymptotics
and far from the bifurcation set, the cuspoid canonical integrals are approximated by
36.11.2 Ψ K ( x ) = 2 π j = 1 j max ( x ) exp ( i ( Φ K ( t j ( x ) ; x ) + 1 4 π ( - 1 ) j + K + 1 ) ) | 2 Φ K ( t j ( x ) ; x ) t 2 | - 1 / 2 ( 1 + o ( 1 ) ) .
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 ) . …
( Φ K ( t j ( x ) ; x ) - Φ K ( t μ ( x ) ; x ) ) = 0 ,
§36.5(ii) Cuspoids
7: 36.10 Differential Equations
Φ s ( U ) ( - i x , - i y ; x ) Ψ ( U ) ( x ) = 0 ,
Φ t ( U ) ( - i x , - i y ; x ) Ψ ( U ) ( x ) = 0 ,
Φ s ( U ) ( s , t ; x ) = s Φ ( U ) ( s , t ; x ) ,
Φ t ( U ) ( s , t ; x ) = t Φ ( U ) ( s , t ; x ) .
8: 36.2 Catastrophes and Canonical Integrals
Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension K
36.2.1 Φ K ( t ; x ) = t K + 2 + m = 1 K x m t m .
36.2.4 Ψ K ( x ) = - exp ( i Φ K ( t ; x ) ) d t .
36.2.10 Ψ K ( x ; k ) = k - exp ( i k Φ K ( t ; x ) ) d t , k > 0 .
9: 36.7 Zeros
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 ) .
10: Bibliography K
  • N. P. Kirk, J. N. L. Connor, and C. A. Hobbs (2000) An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives. Computer Physics Comm. 132 (1-2), pp. 142–165.