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1: 36.2 Catastrophes and Canonical Integrals
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: 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.

  • Chapters 8, 20, 36

    Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

  • 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.

  • References

    Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

  • 3: 36.4 Bifurcation Sets
    Critical Points for Umbilics
    K = 1 , fold bifurcation set: … Hyperbolic umbilic bifurcation set (codimension three): … Hyperbolic umbilic cusp line (rib): …
    §36.4(ii) Visualizations
    4: 36.5 Stokes Sets
    §36.5(ii) Cuspoids
    §36.5(iii) Umbilics
    Elliptic Umbilic Stokes Set (Codimension three)
    Hyperbolic Umbilic Stokes Set (Codimension three)
    §36.5(iv) Visualizations