elliptic%20umbilic%20canonical%20integral

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1: 36.2 Catastrophes and Canonical Integrals

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36.2.28 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-z\right)}=2\pi \sqrt{\frac{\pi z}{27}}\exp \left(\frac{2}{27}\mathrm{i}{z}^3\right)\left(J_{-1/6}\left(\frac{2}{27}{z}^3\right)+\mathrm{i}{J}_{1/6}\left(\frac{2}{27}{z}^3\right)\right),$$ $z\ge 0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit and $z$: real parameter

Referenced by:

Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/36.2.E28

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. For the proof see Berry and Howls (2010).

See also:

Annotations for §36.2(iv), §36.2 and Ch.36

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36.2.29 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,-z\right)}=\frac{2^{1/3}}{\sqrt{3}}\exp \left(\frac{1}{27}\mathrm{i}{z}^3\right){\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-\frac{z}{2^{2/3}}\right),$$ .

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Symbols:

$\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit and $z$: real parameter

Referenced by:

Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/36.2.E29

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. For the proof see Berry and Howls (2010).

See also:

Annotations for §36.2(iv), §36.2 and Ch.36

2: Errata

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22.14.16 $${\int }_0^{K\left(k\right)}\ln \left(\mathrm{sn}(t,k)\right)dt=-\frac{\pi }{4}{K}^{\prime }\left(k\right)-\frac{1}{2}K\left(k\right)\ln k,$$

22.14.17 $${\int }_0^{K\left(k\right)}\ln \left(\mathrm{cn}(t,k)\right)dt=-\frac{\pi }{4}{K}^{\prime }\left(k\right)+\frac{1}{2}K\left(k\right)\ln \left(k^{\prime }/k\right)$$

Originally, a factor of $\pi$ was missing from the terms containing the $-\frac{1}{4}{K}^{\prime }\left(k\right)$.

Reported by Fred Hucht on 2020-08-06

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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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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).

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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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3: 36.4 Bifurcation Sets

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Critical Points for Umbilics

… ► $K=1$, fold bifurcation set: … ►Elliptic umbilic bifurcation set (codimension three): for fixed $z$, the section of the bifurcation set is a three-cusped astroid … ►Elliptic umbilic cusp lines (ribs): … ►

§36.4(ii) Visualizations

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4: 36.5 Stokes Sets

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§36.5(ii) Cuspoids

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§36.5(iii) Umbilics

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Elliptic Umbilic Stokes Set (Codimension three)

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Hyperbolic Umbilic Stokes Set (Codimension three)

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§36.5(iv) Visualizations

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5: 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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The elliptic umbilic diffraction catastrophe. Phil. Trans. Roy. Soc. Ser. A 291 (1382), pp. 453–484.

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External Links:

FullText

Referenced by:

§36.15(iv), §36.2(i), §36.7(ii), §36.7(iii).

Encodings:

BibTeX

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Numerical calculation of elliptic integrals and elliptic functions. III. Numer. Math. 13 (4), pp. 305–315.

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Notes:

Algol 60 programs for the incomplete elliptic integral of the third kind, and for a general complete elliptic integral, are included.

External Links:

ISSN 0029-599X, Document, MathReview, ZentralBlatt

Referenced by:

§19.2(iii), §19.36(ii), §19.36(ii), §19.39(ii), §19.39(iii), R. Bulirsch (1965a).

Encodings:

BibTeX

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Numerical calculation of elliptic integrals and elliptic functions. Numer. Math. 7 (1), pp. 78–90.

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Notes:

Algol 60 programs are included for real and complex elliptic integrals, and for Jacobian elliptic functions of real argument.

External Links:

ISSN 0029-599X, Document, MathReview (M. Feliciano), ZentralBlatt

Referenced by:

§19.1, §19.2(iii), §19.36(ii), §19.39(iii), §22.22(ii).

Encodings:

BibTeX

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On a generalization of Barton’s integral and related integrals of complete elliptic integrals. Math. Proc. Cambridge Philos. Soc. 101 (1), pp. 1–5.

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External Links:

ISSN 0305-0041, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§19.13(i).

Encodings:

BibTeX

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