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hyperbolic umbilic canonical integral

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1: 36.2 Catastrophes and Canonical Integrals
Canonical Integrals
Ψ ( H ) ( 0 ) = 1 3 Γ 2 ( 1 3 ) = 2.39224 .
36.2.21 Ψ ( H ) ( x , y , 0 ) = 4 π 2 3 2 / 3 Ai ( x 3 1 / 3 ) Ai ( y 3 1 / 3 ) .
36.2.27 Ψ ( H ) ( x , y , z ) = Ψ ( H ) ( y , x , z ) .
36.2.29 Ψ ( H ) ( 0 , 0 , z ) = Ψ ( H ) ( 0 , 0 , - z ) ¯ = 2 1 / 3 3 exp ( 1 27 i z 3 ) Ψ ( E ) ( 0 , 0 , - z 2 2 / 3 ) , - < z < .
2: 36.3 Visualizations of Canonical Integrals
Figure 36.3.9: Modulus of hyperbolic umbilic canonical integral function | Ψ ( H ) ( x , y , 0 ) | . …
Figure 36.3.18: Phase of hyperbolic umbilic canonical integral ph Ψ ( H ) ( x , y , 0 ) . …
Figure 36.3.19: Phase of hyperbolic umbilic canonical integral ph Ψ ( H ) ( x , y , 1 ) . …
Figure 36.3.20: Phase of hyperbolic umbilic canonical integral ph Ψ ( H ) ( x , y , 2 ) . …
Figure 36.3.21: Phase of hyperbolic umbilic canonical integral ph Ψ ( H ) ( x , y , 3 ) . …
3: 36.10 Differential Equations
In terms of the normal forms (36.2.2) and (36.2.3), the Ψ ( U ) ( x ) satisfy the following operator equations …
36.10.14 3 ( 2 Ψ ( E ) x 2 - 2 Ψ ( E ) y 2 ) + 2 i z Ψ ( E ) x - x Ψ ( E ) = 0 .
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: 36.9 Integral Identities
§36.9 Integral Identities
36.9.8 | Ψ ( H ) ( x , y , z ) | 2 = 8 π 2 ( 2 9 ) 1 / 3 - - Ai ( ( 4 3 ) 1 / 3 ( x + z v + 3 u 2 ) ) Ai ( ( 4 3 ) 1 / 3 ( y + z u + 3 v 2 ) ) d u d v .
6: 36.11 Leading-Order Asymptotics
§36.11 Leading-Order Asymptotics
36.11.8 Ψ ( H ) ( 0 , 0 , z ) = 2 π z ( 1 - i 3 exp ( 1 27 i z 3 ) + o ( 1 ) ) , z ± .
7: 36.8 Convergent Series Expansions
§36.8 Convergent Series Expansions
36.8.3 3 2 / 3 4 π 2 Ψ ( H ) ( 3 1 / 3 x ) = Ai ( x ) Ai ( y ) n = 0 ( - 3 - 1 / 3 i z ) n c n ( x ) c n ( y ) n ! + Ai ( x ) Ai ( y ) n = 2 ( - 3 - 1 / 3 i z ) n c n ( x ) d n ( y ) n ! + Ai ( x ) Ai ( y ) n = 2 ( - 3 - 1 / 3 i z ) n d n ( x ) c n ( y ) n ! + Ai ( x ) Ai ( y ) n = 1 ( - 3 - 1 / 3 i z ) n d n ( x ) d n ( y ) n ! ,
8: Errata
  • Figures 36.3.9, 36.3.10, 36.3.11, 36.3.12


    Scales were corrected in all figures. The interval - 8.4 x - y 2 8.4 was replaced by - 12.0 x - y 2 12.0 and - 12.7 x + y 2 4.2 replaced by - 18.0 x + y 2 6.0 . All plots and interactive visualizations were regenerated to improve image quality.

    See accompanying text See accompanying text
    (a) Density plot. (b) 3D plot.

    Figure 36.3.9: Modulus of hyperbolic umbilic canonical integral function | Ψ ( H ) ( x , y , 0 ) | .

    See accompanying text See accompanying text
    (a) Density plot. (b) 3D plot.

    Figure 36.3.10: Modulus of hyperbolic umbilic canonical integral function | Ψ ( H ) ( x , y , 1 ) | .

    See accompanying text See accompanying text
    (a) Density plot. (b) 3D plot.

    Figure 36.3.11: Modulus of hyperbolic umbilic canonical integral function | Ψ ( H ) ( x , y , 2 ) | .

    See accompanying text See accompanying text
    (a) Density plot. (b) 3D plot.

    Figure 36.3.12: Modulus of hyperbolic umbilic canonical integral function | Ψ ( H ) ( x , y , 3 ) | .

    Reported 2016-09-12 by Dan Piponi.

  • Figures 36.3.18, 36.3.19, 36.3.20, 36.3.21


    The scaling error reported on 2016-09-12 by Dan Piponi also applied to contour and density plots for the phase of the hyperbolic umbilic canonical integrals. Scales were corrected in all figures. The interval - 8.4 x - y 2 8.4 was replaced by - 12.0 x - y 2 12.0 and - 12.7 x + y 2 4.2 replaced by - 18.0 x + y 2 6.0 . All plots and interactive visualizations were regenerated to improve image quality.

    See accompanying text See accompanying text
    (a) Contour plot. (b) Density plot.

    Figure 36.3.18: Phase of hyperbolic umbilic canonical integral ph Ψ ( H ) ( x , y , 0 ) .

    See accompanying text See accompanying text
    (a) Contour plot. (b) Density plot.

    Figure 36.3.19: Phase of hyperbolic umbilic canonical integral ph Ψ ( H ) ( x , y , 1 ) .

    See accompanying text See accompanying text
    (a) Contour plot. (b) Density plot.

    Figure 36.3.20: Phase of hyperbolic umbilic canonical integral ph Ψ ( H ) ( x , y , 2 ) .

    See accompanying text See accompanying text
    (a) Contour plot. (b) Density plot.

    Figure 36.3.21: Phase of hyperbolic umbilic canonical integral ph Ψ ( H ) ( x , y , 3 ) .

    Reported 2016-09-28.

  • Equation (36.10.14)

    36.10.14
    3 ( 2 Ψ ( E ) x 2 - 2 Ψ ( E ) y 2 ) + 2 i z Ψ ( E ) x - x Ψ ( E ) = 0

    Originally this equation appeared with Ψ ( H ) x in the second term, rather than Ψ ( E ) x .

    Reported 2010-04-02.

  • 9: 36.6 Scaling Relations
    §36.6 Scaling Relations
    10: 36.7 Zeros
    §36.7(iv) Swallowtail and Hyperbolic Umbilic Canonical Integrals
    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 ) .