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1: 36.6 Scaling Relations
ā–ŗ
cuspoids ā¢ š² ā¢ ( k ) = ( x 1 ā¢ k Ī³ 1 ā¢ K , x 2 ā¢ k Ī³ 2 ā¢ K , , x K ā¢ k Ī³ K ā¢ K ) ,
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cuspoids ā¢ Ī² K = K 2 ā¢ ( K + 2 ) ,
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cuspoids ā¢ Ī³ m ā¢ K = 1 m K + 2 ,
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cuspoids ā¢ Ī³ K = m = 1 K Ī³ m ā¢ K = K ā¢ ( K + 3 ) 2 ā¢ ( K + 2 ) ,
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Table 36.6.1: Special cases of scaling exponents for cuspoids.
ā–ŗ ā–ŗā–ŗ
singularity K Ī² K Ī³ 1 ā¢ K Ī³ 2 ā¢ K Ī³ 3 ā¢ K Ī³ K
ā–ŗ
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 ā” ( t ; š² ) by relating f ā” ( u ; š² ) to the normal form (36.2.1) of Ī¦ K ā” ( t ; š± ) in the following way: …with the K + 1 functions A ā” ( š² ) and š± ā” ( š² ) determined by correspondence of the K + 1 critical points of f and Ī¦ K . …where t j ā” ( š± ) , 1 j K + 1 , are the critical points of Ī¦ 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
ā–ŗ ā–ŗ
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 Ī¦ K ā” ( t ; š± ) ; umbilic catastrophes with codimension three Ī¦ ( E ) ā” ( s , t ; š± ) , Ī¦ ( H ) ā” ( s , t ; š± ) ; canonical integrals ĪØ K ā” ( š± ) , ĪØ ( E ) ā” ( š± ) , ĪØ ( H ) ā” ( š± ) ; diffraction catastrophes ĪØ K ā” ( š± ; k ) , ĪØ ( E ) ā” ( š± ; k ) , ĪØ ( H ) ā” ( š± ; k ) generated by the catastrophes. …
5: 36.5 Stokes Sets
ā–ŗStokes sets are surfaces (codimension one) in š± space, across which ĪØ K ā” ( š± ; k ) or ĪØ ( U ) ā” ( š± ; k ) acquires an exponentially-small asymptotic contribution (in k ), associated with a complex critical point of Ī¦ K or Ī¦ ( U ) . … ā–ŗ
ā” ( Ī¦ K ā” ( t j ā” ( š± ) ; š± ) Ī¦ K ā” ( t Ī¼ ā” ( š± ) ; š± ) ) = 0 ,
ā–ŗ
§36.5(ii) Cuspoids
6: 36.10 Differential Equations
ā–ŗ ā–ŗ
Ī¦ s ( U ) ā¢ ( i ā¢ x , i ā¢ y ; š± ) ā¢ ĪØ ( U ) ā” ( š± ) = 0 ,
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Ī¦ t ( U ) ā¢ ( i ā¢ x , i ā¢ y ; š± ) ā¢ ĪØ ( U ) ā” ( š± ) = 0 ,
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Ī¦ s ( U ) ā” ( s , t ; š± ) = s ā” Ī¦ ( U ) ā” ( s , t ; š± ) ,
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Ī¦ t ( U ) ā” ( s , t ; š± ) = t ā” Ī¦ ( U ) ā” ( s , t ; š± ) .
7: 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 ā” ( š± ) = 2 ā¢ Ļ€ ā¢ j = 1 j max ā¢ ( š± ) exp ā” ( i ā¢ ( Ī¦ K ā” ( t j ā” ( š± ) ; š± ) + 1 4 ā¢ Ļ€ ā¢ ( 1 ) j + K + 1 ) ) ā¢ | 2 Ī¦ K ā” ( t j ā” ( š± ) ; š± ) t 2 | 1 / 2 ā¢ ( 1 + o ā” ( 1 ) ) .
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 Ī¦ K ā” ( t ; š± ) = t K + 2 + m = 1 K x m ā¢ t m .
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36.2.4 ĪØ K ā” ( š± ) = exp ā” ( i ā¢ Ī¦ K ā” ( t ; š± ) ) ā¢ d t .
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36.2.10 ĪØ K ā” ( š± ; k ) = k ā¢ exp ā” ( i ā¢ k ā¢ Ī¦ K ā” ( t ; š± ) ) ā¢ 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 Ī¦ 3 and Nye (2006) for Ī¦ ( H ) .
10: Bibliography K
ā–ŗ
  • 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.