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1: 36.3 Visualizations of Canonical Integrals
See accompanying text
See accompanying text
Figure 36.3.1: Modulus of Pearcey integral | Ψ 2 ( x , y ) | . Magnify 3D Help
Figure 36.3.2: Modulus of swallowtail canonical integral function | Ψ 3 ( x , y , 3 ) | .
Figure 36.3.3: Modulus of swallowtail canonical integral function | Ψ 3 ( x , y , 0 ) | .
Figure 36.3.4: Modulus of swallowtail canonical integral function | Ψ 3 ( x , y , 3 ) | .
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.

  • 3: 18.39 Applications in the Physical Sciences
    A relativistic treatment becoming necessary as Z becomes large as corrections to the non-relativistic Schrödinger picture are of approximate order ( α Z ) 2 ( Z / 137 ) 2 , α being the dimensionless fine structure constant e 2 / ( 4 π ε 0 c ) , where c is the speed of light. …
    4: 36.5 Stokes Sets
    The Stokes set consists of the rays ph x = ± 2 π / 3 in the complex x -plane. … For z < 0 , there are two solutions u , provided that | Y | > ( 2 5 ) 1 / 2 . … This consists of three separate cusp-edged sheets connected to the cusp-edged sheets of the bifurcation set, and related by rotation about the z -axis by 2 π / 3 . …
    §36.5(iv) Visualizations
    See accompanying text
    Figure 36.5.8: Sheets of the Stokes surface for the elliptic umbilic catastrophe (colored and with mesh) and the bifurcation set (gray). Magnify