36 Integrals with Coalescing SaddlesProperties36.2 Catastrophes and Canonical Integrals36.4 Bifurcation Sets

§36.3 Visualizations of Canonical Integrals

ⓘ

Referenced by:

Erratum (V1.0.14) for Section 36.3

Permalink:

http://dlmf.nist.gov/36.3

See also:

Annotations for Ch.36

Contents

1. §36.3(i) Canonical Integrals: Modulus
2. §36.3(ii) Canonical Integrals: Phase

§36.3(i) Canonical Integrals: Modulus

ⓘ

Keywords:

canonical integrals, visualizations of modulus

Notes:

These figures were generated by the authors.

Permalink:

http://dlmf.nist.gov/36.3.i

See also:

Annotations for §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $y$: real parameter and $x$: real parameter

Keywords:

Pearcey integral, pictures of modulus

Referenced by:

Erratum (V1.0.14) for Section 36.3

Permalink:

http://dlmf.nist.gov/36.3.F1

Encodings:

Vizualization, pdf, pdf, png, png

See also:

Annotations for §36.3(i), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $y$: real parameter and $x$: real parameter

Keywords:

pictures of modulus, swallowtail canonical integral

Referenced by:

Erratum (V1.0.14) for Section 36.3

Permalink:

http://dlmf.nist.gov/36.3.F2

Encodings:

Vizualization, pdf, pdf, png, png

See also:

Annotations for §36.3(i), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $y$: real parameter and $x$: real parameter

Keywords:

pictures of modulus, swallowtail canonical integral

Referenced by:

Erratum (V1.0.14) for Section 36.3

Permalink:

http://dlmf.nist.gov/36.3.F3

Encodings:

Vizualization, pdf, pdf, png, png

See also:

Annotations for §36.3(i), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $y$: real parameter and $x$: real parameter

Keywords:

pictures of modulus, swallowtail canonical integral

Referenced by:

Erratum (V1.0.14) for Section 36.3

Permalink:

http://dlmf.nist.gov/36.3.F4

Encodings:

Vizualization, pdf, pdf, png, png

See also:

Annotations for §36.3(i), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $y$: real parameter and $x$: real parameter

Keywords:

pictures of modulus, swallowtail canonical integral

Referenced by:

Erratum (V1.0.14) for Section 36.3

Permalink:

http://dlmf.nist.gov/36.3.F5

Encodings:

Vizualization, pdf, pdf, png, png

See also:

Annotations for §36.3(i), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $y$: real parameter and $x$: real parameter

Keywords:

elliptic umbilic canonical integral, pictures of modulus

Referenced by:

Erratum (V1.0.14) for Section 36.3

Permalink:

http://dlmf.nist.gov/36.3.F6

Encodings:

Vizualization, pdf, pdf, png, png

See also:

Annotations for §36.3(i), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $y$: real parameter and $x$: real parameter

Keywords:

elliptic umbilic canonical integral, pictures of modulus

Referenced by:

Erratum (V1.0.14) for Section 36.3

Permalink:

http://dlmf.nist.gov/36.3.F7

Encodings:

Vizualization, pdf, pdf, png, png

See also:

Annotations for §36.3(i), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $y$: real parameter and $x$: real parameter

Keywords:

elliptic umbilic canonical integral, pictures of modulus

Referenced by:

Erratum (V1.0.14) for Section 36.3

Permalink:

http://dlmf.nist.gov/36.3.F8

Encodings:

Vizualization, pdf, pdf, png, png

See also:

Annotations for §36.3(i), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $y$: real parameter and $x$: real parameter

Keywords:

hyperbolic umbilic canonical integral, pictures of modulus

Referenced by:

Erratum (V1.0.14) for Figures 36.3.9, 36.3.10, 36.3.11, 36.3.12

Permalink:

http://dlmf.nist.gov/36.3.F9

Encodings:

Vizualization, pdf, pdf, png, png

Errata (effective with 1.0.14):

Scales were corrected. 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 $-18.0\le \frac{x+y}{\sqrt{2}}\le 6.0.$

Reported 2016-09-12 by Dan  Piponi

See also:

Annotations for §36.3(i), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $y$: real parameter and $x$: real parameter

Keywords:

hyperbolic umbilic canonical integral, pictures of modulus

Referenced by:

Erratum (V1.0.14) for Figures 36.3.9, 36.3.10, 36.3.11, 36.3.12

Permalink:

http://dlmf.nist.gov/36.3.F10

Encodings:

Vizualization, pdf, pdf, png, png

Errata (effective with 1.0.14):

Scales were corrected. 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 $-18.0\le \frac{x+y}{\sqrt{2}}\le 6.0.$

Reported 2016-09-12 by Dan  Piponi

See also:

Annotations for §36.3(i), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $y$: real parameter and $x$: real parameter

Keywords:

hyperbolic umbilic canonical integral, pictures of modulus

Referenced by:

Erratum (V1.0.14) for Figures 36.3.9, 36.3.10, 36.3.11, 36.3.12

Permalink:

http://dlmf.nist.gov/36.3.F11

Encodings:

Vizualization, pdf, pdf, png, png

Errata (effective with 1.0.14):

Scales were corrected. 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 $-18.0\le \frac{x+y}{\sqrt{2}}\le 6.0.$

Reported 2016-09-12 by Dan  Piponi

See also:

Annotations for §36.3(i), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $y$: real parameter and $x$: real parameter

Keywords:

hyperbolic umbilic canonical integral, pictures of modulus

Referenced by:

Erratum (V1.0.14) for Figures 36.3.9, 36.3.10, 36.3.11, 36.3.12

Permalink:

http://dlmf.nist.gov/36.3.F12

Encodings:

Vizualization, pdf, pdf, png, png

Errata (effective with 1.0.14):

Scales were corrected. 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 $-18.0\le \frac{x+y}{\sqrt{2}}\le 6.0.$

Reported 2016-09-12 by Dan  Piponi

See also:

Annotations for §36.3(i), §36.3 and Ch.36

§36.3(ii) Canonical Integrals: Phase

ⓘ

Keywords:

canonical integrals, visualizations of phase

Notes:

These figures were generated by the authors.

Permalink:

http://dlmf.nist.gov/36.3.ii

See also:

Annotations for §36.3 and Ch.36

In Figure 36.3.13(a) points of confluence of phase contours are zeros of ${\mathrm{\Psi }}_2\left(x,y\right)$; similarly for other contour plots in this subsection. In Figure 36.3.13(b) points of confluence of all colors are zeros of ${\mathrm{\Psi }}_2\left(x,y\right)$; similarly for other density plots in this subsection.

(a) Contour plot, at intervals of $\pi /4$.	(b) Density plot.

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $y$: real parameter and $x$: real parameter

Keywords:

Pearcey integral, pictures of phase

Referenced by:

§36.3(ii), Erratum (V1.0.14) for Section 36.3

Permalink:

http://dlmf.nist.gov/36.3.F13

Encodings:

pdf, pdf, png, png

See also:

Annotations for §36.3(ii), §36.3 and Ch.36

(a) $\mathrm{ph}{\mathrm{\Psi }}_3\left(x,y,3\right)$.	(b) $\mathrm{ph}{\mathrm{\Psi }}_3\left(x,y,0\right)$.
(c) $\mathrm{ph}{\mathrm{\Psi }}_3\left(x,y,-3\right)$.	(d) $\mathrm{ph}{\mathrm{\Psi }}_3\left(x,y,-7.5\right)$.

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{ph}$: phase, $y$: real parameter and $x$: real parameter

Keywords:

pictures of phase, swallowtail canonical integral

Referenced by:

Erratum (V1.0.14) for Section 36.3

Permalink:

http://dlmf.nist.gov/36.3.F14

Encodings:

pdf, pdf, pdf, pdf, png, png, png, png

See also:

Annotations for §36.3(ii), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\mathrm{ph}$: phase, $y$: real parameter and $x$: real parameter

Keywords:

elliptic umbilic canonical integral, pictures of phase

Referenced by:

Erratum (V1.0.14) for Section 36.3

Permalink:

http://dlmf.nist.gov/36.3.F15

Encodings:

pdf, pdf, png, png

See also:

Annotations for §36.3(ii), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\mathrm{ph}$: phase, $y$: real parameter and $x$: real parameter

Keywords:

elliptic umbilic canonical integral, pictures of phase

Referenced by:

Erratum (V1.0.14) for Section 36.3

Permalink:

http://dlmf.nist.gov/36.3.F16

Encodings:

pdf, pdf, png, png

See also:

Annotations for §36.3(ii), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\mathrm{ph}$: phase, $y$: real parameter and $x$: real parameter

Keywords:

elliptic umbilic canonical integral, pictures of phase

Referenced by:

Erratum (V1.0.14) for Section 36.3

Permalink:

http://dlmf.nist.gov/36.3.F17

Encodings:

pdf, pdf, png, png

See also:

Annotations for §36.3(ii), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{ph}$: phase, $y$: real parameter and $x$: real parameter

Keywords:

hyperbolic umbilic canonical integral, pictures of phase

Referenced by:

Erratum (V1.0.14) for Figures 36.3.18, 36.3.19, 36.3.20, 36.3.21

Permalink:

http://dlmf.nist.gov/36.3.F18

Encodings:

pdf, pdf, png, png

Errata (effective with 1.0.14):

Scales were corrected. 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 $-18.0\le \frac{x+y}{\sqrt{2}}\le 6.0.$

See also:

Annotations for §36.3(ii), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{ph}$: phase, $y$: real parameter and $x$: real parameter

Keywords:

hyperbolic umbilic canonical integral, pictures of phase

Referenced by:

Erratum (V1.0.14) for Figures 36.3.18, 36.3.19, 36.3.20, 36.3.21

Permalink:

http://dlmf.nist.gov/36.3.F19

Encodings:

pdf, pdf, png, png

Errata (effective with 1.0.14):

Scales were corrected. 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 $-18.0\le \frac{x+y}{\sqrt{2}}\le 6.0.$

See also:

Annotations for §36.3(ii), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{ph}$: phase, $y$: real parameter and $x$: real parameter

Keywords:

hyperbolic umbilic canonical integral, pictures of phase

Referenced by:

Erratum (V1.0.14) for Figures 36.3.18, 36.3.19, 36.3.20, 36.3.21

Permalink:

http://dlmf.nist.gov/36.3.F20

Encodings:

pdf, pdf, png, png

Errata (effective with 1.0.14):

Scales were corrected. 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 $-18.0\le \frac{x+y}{\sqrt{2}}\le 6.0.$

See also:

Annotations for §36.3(ii), §36.3 and Ch.36

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

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{ph}$: phase, $y$: real parameter and $x$: real parameter

Keywords:

hyperbolic umbilic canonical integral, pictures of phase

Referenced by:

Erratum (V1.0.14) for Figures 36.3.18, 36.3.19, 36.3.20, 36.3.21

Permalink:

http://dlmf.nist.gov/36.3.F21

Encodings:

pdf, pdf, png, png

Errata (effective with 1.0.14):

Scales were corrected. 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 $-18.0\le \frac{x+y}{\sqrt{2}}\le 6.0.$

See also:

Annotations for §36.3(ii), §36.3 and Ch.36

36.2 Catastrophes and Canonical Integrals36.4 Bifurcation Sets

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