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1: 36.6 Scaling Relations
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cuspoids ⁢ š² ⁢ ( k ) = ( x 1 ⁢ k γ 1 ⁢ K , x 2 ⁢ k γ 2 ⁢ K , , x K ⁢ k γ K ⁢ K ) ,
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cuspoids ⁢ β K = K 2 ⁢ ( K + 2 ) ,
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cuspoids ⁢ γ m ⁢ K = 1 m K + 2 ,
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cuspoids ⁢ γ K = m = 1 K γ m ⁢ K = K ⁢ ( K + 3 ) 2 ⁢ ( K + 2 ) ,
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Table 36.6.1: Special cases of scaling exponents for cuspoids.
ā–ŗ ā–ŗā–ŗ
singularity K β K γ 1 ⁢ K γ 2 ⁢ K γ 3 ⁢ K γ K
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2: 36.12 Uniform Approximation of Integrals
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§36.12(i) General Theory for Cuspoids
ā–ŗIn the cuspoid case (one integration variable) … ā–ŗDefine a mapping u ⁔ ( t ; š² ) by relating f ⁔ ( u ; š² ) to the normal form (36.2.1) of Φ K ⁔ ( t ; š± ) in the following way: …with the K + 1 functions A ⁔ ( š² ) and š± ⁔ ( š² ) determined by correspondence of the K + 1 critical points of f and Φ K . …where t j ⁔ ( š± ) , 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
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Critical Points for Cuspoids
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Bifurcation (Catastrophe) Set for Cuspoids
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4: 36.1 Special Notation
ā–ŗThe main functions covered in this chapter are cuspoid catastrophes Φ K ⁔ ( t ; š± ) ; umbilic catastrophes with codimension three Φ ( E ) ⁔ ( s , t ; š± ) , Φ ( H ) ⁔ ( s , t ; š± ) ; canonical integrals ĪØ K ⁔ ( š± ) , ĪØ ( E ) ⁔ ( š± ) , ĪØ ( H ) ⁔ ( š± ) ; diffraction catastrophes ĪØ K ⁔ ( š± ; k ) , ĪØ ( E ) ⁔ ( š± ; k ) , ĪØ ( H ) ⁔ ( š± ; k ) generated by the catastrophes. …
5: 36.11 Leading-Order Asymptotics
ā–ŗand far from the bifurcation set, the cuspoid canonical integrals are approximated by ā–ŗ
36.11.2 ĪØ K ⁔ ( š± ) = 2 ⁢ Ļ€ ⁢ j = 1 j max ⁢ ( š± ) exp ⁔ ( i ⁢ ( Φ K ⁔ ( t j ⁔ ( š± ) ; š± ) + 1 4 ⁢ Ļ€ ⁢ ( 1 ) j + K + 1 ) ) ⁢ | 2 Φ K ⁔ ( t j ⁔ ( š± ) ; š± ) t 2 | 1 / 2 ⁢ ( 1 + o ⁔ ( 1 ) ) .
6: 36.5 Stokes Sets
ā–ŗStokes sets are surfaces (codimension one) in š± space, across which ĪØ K ⁔ ( š± ; k ) or ĪØ ( U ) ⁔ ( š± ; k ) acquires an exponentially-small asymptotic contribution (in k ), associated with a complex critical point of Φ K or Φ ( U ) . … ā–ŗ
⁔ ( Φ K ⁔ ( t j ⁔ ( š± ) ; š± ) Φ K ⁔ ( t μ ⁔ ( š± ) ; š± ) ) = 0 ,
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§36.5(ii) Cuspoids
7: 36.10 Differential Equations
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Φ s ( U ) ⁢ ( i ⁢ x , i ⁢ y ; š± ) ⁢ ĪØ ( U ) ⁔ ( š± ) = 0 ,
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Φ t ( U ) ⁢ ( i ⁢ x , i ⁢ y ; š± ) ⁢ ĪØ ( U ) ⁔ ( š± ) = 0 ,
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Φ s ( U ) ⁔ ( s , t ; š± ) = s ⁔ Φ ( U ) ⁔ ( s , t ; š± ) ,
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Φ t ( U ) ⁔ ( s , t ; š± ) = t ⁔ Φ ( U ) ⁔ ( s , t ; š± ) .
8: 36.2 Catastrophes and Canonical Integrals
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Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension K
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36.2.1 Φ K ⁔ ( t ; š± ) = t K + 2 + m = 1 K x m ⁢ t m .
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36.2.4 ĪØ K ⁔ ( š± ) = exp ⁔ ( i ⁢ Φ K ⁔ ( t ; š± ) ) ⁢ d t .
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36.2.10 ĪØ K ⁔ ( š± ; k ) = k ⁢ exp ⁔ ( i ⁢ k ⁢ Φ K ⁔ ( t ; š± ) ) ⁢ d t , k > 0 .
9: 36.7 Zeros
ā–ŗThe zeros of these functions are curves in š± = ( x , y , z ) space; see Nye (2007) for Φ 3 and Nye (2006) for Φ ( H ) .
10: Bibliography K
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  • 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.