swallowtail canonical integral

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1: 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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Canonical Integrals

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2: 36.3 Visualizations of Canonical Integrals

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Figure 36.3.2: Modulus of swallowtail canonical integral function $|{\mathrm{\Psi }}_3\left(x,y,3\right)|$.

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Figure 36.3.3: Modulus of swallowtail canonical integral function $|{\mathrm{\Psi }}_3\left(x,y,0\right)|$.

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Figure 36.3.4: Modulus of swallowtail canonical integral function $|{\mathrm{\Psi }}_3\left(x,y,-3\right)|$.

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Figure 36.3.5: Modulus of swallowtail canonical integral function $|{\mathrm{\Psi }}_3\left(x,y,-7.5\right)|$.

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Figure 36.3.14: Density plots of phase of swallowtail canonical integrals.

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3: 36.6 Scaling Relations

§36.6 Scaling Relations

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4: 36.7 Zeros

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§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 ${\mathrm{\Phi }}_3$ and Nye (2006) for ${\mathrm{\Phi }}^{(\mathrm{H})}$.

5: 36.9 Integral Identities

§36.9 Integral Identities

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6: 36.5 Stokes Sets

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§36.5(ii) Cuspoids

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§36.5(iv) Visualizations

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7: 36.11 Leading-Order Asymptotics

§36.11 Leading-Order Asymptotics

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8: 36.8 Convergent Series Expansions

§36.8 Convergent Series Expansions

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9: 36.12 Uniform Approximation of Integrals

… ►For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).

10: 36.10 Differential Equations

… ► $K=3$, swallowtail: …