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36 Integrals with Coalescing SaddlesProperties

§36.3 Visualizations of Canonical Integrals

Contents

§36.3(i) Canonical Integrals: Modulus

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(a) Density plot. (b) 3D plot.
Figure 36.3.1: Modulus of Pearcey integral |Ψ2(x,y)|. Magnify
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(a) Density plot. (b) 3D plot.
Figure 36.3.2: Modulus of swallowtail canonical integral function |Ψ3(x,y,3)|. Magnify
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(a) Density plot. (b) 3D plot.
Figure 36.3.3: Modulus of swallowtail canonical integral function |Ψ3(x,y,0)|. Magnify
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(a) Density plot. (b) 3D plot.
Figure 36.3.4: Modulus of swallowtail canonical integral function |Ψ3(x,y,-3)|. Magnify
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(a) Density plot. (b) 3D plot.
Figure 36.3.5: Modulus of swallowtail canonical integral function |Ψ3(x,y,-7.5)|. Magnify
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(a) Density plot. (b) 3D plot.
Figure 36.3.6: Modulus of elliptic umbilic canonical integral function |Ψ(E)(x,y,0)|. Magnify
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(a) Density plot. (b) 3D plot.
Figure 36.3.7: Modulus of elliptic umbilic canonical integral function |Ψ(E)(x,y,2)|. Magnify
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(a) Density plot. (b) 3D plot.
Figure 36.3.8: Modulus of elliptic umbilic canonical integral function |Ψ(E)(x,y,4)|. Magnify
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(a) Density plot. (b) 3D plot.
Figure 36.3.9: Modulus of hyperbolic umbilic canonical integral function |Ψ(H)(x,y,0)|. Magnify
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(a) Density plot. (b) 3D plot.
Figure 36.3.10: Modulus of hyperbolic umbilic canonical integral function |Ψ(H)(x,y,1)|. Magnify
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(a) Density plot. (b) 3D plot.
Figure 36.3.11: Modulus of hyperbolic umbilic canonical integral function |Ψ(H)(x,y,2)|. Magnify
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(a) Density plot. (b) 3D plot.
Figure 36.3.12: Modulus of hyperbolic umbilic canonical integral function |Ψ(H)(x,y,3)|. Magnify

§36.3(ii) Canonical Integrals: Phase

In Figure 36.3.13(a) points of confluence of phase contours are zeros of Ψ2(x,y); similarly for other contour plots in this subsection. In Figure 36.3.13(b) points of confluence of all colors are zeros of Ψ2(x,y); similarly for other density plots in this subsection.

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(a) Contour plot, at intervals of π/4. (b) Density plot.
Figure 36.3.13: Phase of Pearcey integral phΨ2(x,y). Magnify
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(a) phΨ3(x,y,3). (b) phΨ3(x,y,0).
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(c) phΨ3(x,y,-3). (d) phΨ3(x,y,-7.5).
Figure 36.3.14: Density plots of phase of swallowtail canonical integrals. Magnify
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(a) Contour plot. (b) Density plot.
Figure 36.3.15: Phase of elliptic umbilic canonical integral phΨ(E)(x,y,0). Magnify
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(a) Contour plot. (b) Density plot.
Figure 36.3.16: Phase of elliptic umbilic canonical integral phΨ(E)(x,y,2). Magnify
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(a) Contour plot. (b) Density plot.
Figure 36.3.17: Phase of elliptic umbilic canonical integral phΨ(E)(x,y,4). Magnify
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(a) Contour plot. (b) Density plot.
Figure 36.3.18: Phase of hyperbolic umbilic canonical integral phΨ(H)(x,y,0). Magnify
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(a) Contour plot. (b) Density plot.
Figure 36.3.19: Phase of hyperbolic umbilic canonical integral phΨ(H)(x,y,1). Magnify
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(a) Contour plot. (b) Density plot.
Figure 36.3.20: Phase of hyperbolic umbilic canonical integral phΨ(H)(x,y,2). Magnify
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(a) Contour plot. (b) Density plot.
Figure 36.3.21: Phase of hyperbolic umbilic canonical integral phΨ(H)(x,y,3). Magnify