canonical integrals

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11: 36.12 Uniform Approximation of Integrals

§36.12 Uniform Approximation of Integrals

… ►The canonical integrals (36.2.4) provide a basis for uniform asymptotic approximations of oscillatory integrals. … ►This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes ${\mathrm{\Psi }}_K(𝐱;k)$ in (36.2.10) away from $𝐱=\mathrm{𝟎}$, in terms of canonical integrals ${\mathrm{\Psi }}_J\left(\xi (𝐱;k)\right)$ for . For example, the diffraction catastrophe ${\mathrm{\Psi }}_2(x,y;k)$ defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function ${\mathrm{\Psi }}_1\left(\xi (x,y;k)\right)$ when $k$ is large, provided that $x$ and $y$ are not small. … ►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).

12: Errata

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

(a) Density plot.	(b) 3D plot.

Figure 36.3.9: Modulus of hyperbolic umbilic canonical integral function $|{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,0\right)|$.

(a) Density plot.	(b) 3D plot.

Figure 36.3.10: Modulus of hyperbolic umbilic canonical integral function $|{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,1\right)|$.

(a) Density plot.	(b) 3D plot.

Figure 36.3.11: Modulus of hyperbolic umbilic canonical integral function $|{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,2\right)|$.

(a) Density plot.	(b) 3D plot.

Figure 36.3.12: Modulus of hyperbolic umbilic canonical integral function $|{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,3\right)|$.

Reported 2016-09-12 by Dan Piponi.

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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 $-8.4\le \frac{x-y}{\sqrt{2}}\le 8.4$ was replaced by $-12.0\le \frac{x-y}{\sqrt{2}}\le 12.0$ and $-12.7\le \frac{x+y}{\sqrt{2}}\le 4.2$ replaced by $-18.0\le \frac{x+y}{\sqrt{2}}\le 6.0$. All plots and interactive visualizations were regenerated to improve image quality.

(a) Contour plot.	(b) Density plot.

Figure 36.3.18: Phase of hyperbolic umbilic canonical integral $\mathrm{ph}{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,0\right)$.

(a) Contour plot.	(b) Density plot.

Figure 36.3.19: Phase of hyperbolic umbilic canonical integral $\mathrm{ph}{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,1\right)$.

(a) Contour plot.	(b) Density plot.

Figure 36.3.20: Phase of hyperbolic umbilic canonical integral $\mathrm{ph}{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,2\right)$.

(a) Contour plot.	(b) Density plot.

Figure 36.3.21: Phase of hyperbolic umbilic canonical integral $\mathrm{ph}{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,3\right)$.

Reported 2016-09-28.

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36.10.14 $$3\left(\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial x}^2}-\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial y}^2}\right)+2\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x}-x{\mathrm{\Psi }}^{(\mathrm{E})}=0$$

Originally this equation appeared with $\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial x}$ in the second term, rather than $\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x}$.

Reported 2010-04-02.

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13: 36.15 Methods of Computation

… ►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of $\mathrm{\Phi }$, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. … ►For numerical solution of partial differential equations satisfied by the canonical integrals see Connor et al. (1983).

14: 36.5 Stokes Sets

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§36.5(ii) Cuspoids

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§36.5(iii) Umbilics

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§36.5(iv) Visualizations

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Figure 36.5.8: Sheets of the Stokes surface for the elliptic umbilic catastrophe (colored and with mesh) and the bifurcation set (gray). Magnify

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Figure 36.5.9: Sheets of the Stokes surface for the hyperbolic umbilic catastrophe (colored and with mesh) and the bifurcation set (gray). Magnify

15: 36.4 Bifurcation Sets

… ► $K=1$, fold bifurcation set: …

16: Bibliography U

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Collisions and umbilic catastrophes. The hyperbolic umbilic canonical diffraction integral. Molecular Phys. 50 (6), pp. 1215–1230.

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External Links:

ISSN 0026-8976, Document, MathReview (D. R. J. Chillingworth)

Referenced by:

§36.14(iii).

Encodings:

BibTeX

17: Bibliography C

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A differential equation method for the numerical evaluation of the Airy, Pearcey and swallowtail canonical integrals and their derivatives. Molecular Phys. 48 (6), pp. 1305–1330.

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External Links:

ISSN 0026-8976, Document, MathReview (Peter Giblin)

Referenced by:

§36.10(i), §36.10(ii), §36.15(v), §36.8.

Encodings:

BibTeX

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Evaluation of multidimensional canonical integrals in semiclassical collision theory. Molecular Phys. 26 (6), pp. 1371–1377.

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External Links:

Document

Referenced by:

§36.8.

Encodings:

BibTeX

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18: Bibliography K

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An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives. Computer Physics Comm. 132 (1-2), pp. 142–165.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

§36.15(iii).

Encodings:

BibTeX

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19: Bibliography B

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Overlapping Stokes smoothings: Survival of the error function and canonical catastrophe integrals. Proc. Roy. Soc. London Ser. A 444, pp. 201–216.

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External Links:

ISSN 0962-8444, FullText, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

BibTeX

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20: 27.21 Tables

… ►Bressoud and Wagon (2000, pp. 103–104) supplies tables and graphs that compare $\pi \left(x\right),x/\ln x$, and $\mathrm{li}\left(x\right)$. Glaisher (1940) contains four tables: Table I tabulates, for all $n\le {10}^4$: (a) the canonical factorization of $n$ into powers of primes; (b) the Euler totient $\varphi \left(n\right)$; (c) the divisor function $d\left(n\right)$; (d) the sum $\sigma (n)$ of these divisors. …