36 Integrals with Coalescing Saddles Properties 36.2 Catastrophes and Canonical Integrals 36.4 Bifurcation Sets

§36.3 Visualizations of Canonical Integrals

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

§36.3(i) Canonical Integrals: Modulus
§36.3(ii) Canonical Integrals: Phase

§36.3(i) Canonical Integrals: Modulus

Notes:: These figures were generated by the authors.
Keywords:: canonical integrals
Permalink:: http://dlmf.nist.gov/36.3.i

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

Figure 36.3.1: Modulus of Pearcey integral $|\mathop{\Psi_{{2}}\/}\nolimits\!\left(x,y\right)|$ .

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, : real parameter and : real parameter
Keywords:: Pearcey integral
Permalink:: http://dlmf.nist.gov/36.3.F1
Encodings:: VRML, X3D, pdf, pdf, png, png

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

Figure 36.3.2: Modulus of swallowtail canonical integral function $|\mathop{\Psi_{{3}}\/}\nolimits\!\left(x,y,3\right)|$ .

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, : real parameter and : real parameter
Keywords:: swallowtail canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F2
Encodings:: VRML, X3D, pdf, pdf, png, png

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

Figure 36.3.3: Modulus of swallowtail canonical integral function $|\mathop{\Psi_{{3}}\/}\nolimits\!\left(x,y,0\right)|$ .

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, : real parameter and : real parameter
Keywords:: swallowtail canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F3
Encodings:: VRML, X3D, pdf, pdf, png, png

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

Figure 36.3.4: Modulus of swallowtail canonical integral function $|\mathop{\Psi_{{3}}\/}\nolimits\!\left(x,y,-3\right)|$ .

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, : real parameter and : real parameter
Keywords:: swallowtail canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F4
Encodings:: VRML, X3D, pdf, pdf, png, png

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

Figure 36.3.5: Modulus of swallowtail canonical integral function $|\mathop{\Psi_{{3}}\/}\nolimits\!\left(x,y,-7.5\right)|$ .

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, : real parameter and : real parameter
Keywords:: swallowtail canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F5
Encodings:: VRML, X3D, pdf, pdf, png, png

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

Figure 36.3.6: Modulus of elliptic umbilic canonical integral function $|\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(x,y,0\right)|$ .

Symbols:: $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, : real parameter and : real parameter
Keywords:: elliptic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F6
Encodings:: VRML, X3D, pdf, pdf, png, png

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

Figure 36.3.7: Modulus of elliptic umbilic canonical integral function $|\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(x,y,2\right)|$ .

Symbols:: $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, : real parameter and : real parameter
Keywords:: elliptic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F7
Encodings:: VRML, X3D, pdf, pdf, png, png

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

Figure 36.3.8: Modulus of elliptic umbilic canonical integral function $|\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(x,y,4\right)|$ .

Symbols:: $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, : real parameter and : real parameter
Keywords:: elliptic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F8
Encodings:: VRML, X3D, pdf, pdf, png, png

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

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

Symbols:: $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, : real parameter and : real parameter
Keywords:: hyperbolic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F9
Encodings:: VRML, X3D, pdf, pdf, png, png

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

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

Symbols:: $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, : real parameter and : real parameter
Keywords:: hyperbolic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F10
Encodings:: VRML, X3D, pdf, pdf, png, png

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

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

Symbols:: $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, : real parameter and : real parameter
Keywords:: hyperbolic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F11
Encodings:: VRML, X3D, pdf, pdf, png, png

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

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

Symbols:: $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, : real parameter and : real parameter
Keywords:: hyperbolic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F12
Encodings:: VRML, X3D, pdf, pdf, png, png

§36.3(ii) Canonical Integrals: Phase

Notes:: These figures were generated by the authors.
Keywords:: canonical integrals
Permalink:: http://dlmf.nist.gov/36.3.ii

In Figure 36.3.13(a) points of confluence of phase contours are zeros of $\mathop{\Psi_{{2}}\/}\nolimits\!\left(x,y\right)$ ; similarly for other contour plots in this subsection. In Figure 36.3.13(b) points of confluence of all colors are zeros of $\mathop{\Psi_{{2}}\/}\nolimits\!\left(x,y\right)$ ; similarly for other density plots in this subsection.


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

Figure 36.3.13: Phase of Pearcey integral $\mathop{\mathrm{ph}\/}\nolimits\mathop{\Psi_{{2}}\/}\nolimits\!\left(x,y\right)$ .

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : real parameter and : real parameter
Keywords:: Pearcey integral
Referenced by:: §36.3(ii)
Permalink:: http://dlmf.nist.gov/36.3.F13
Encodings:: pdf, pdf, png, png


(a) $\mathop{\mathrm{ph}\/}\nolimits\mathop{\Psi_{{3}}\/}\nolimits\!\left(x,y,3\right)$ .	(b) $\mathop{\mathrm{ph}\/}\nolimits\mathop{\Psi_{{3}}\/}\nolimits\!\left(x,y,0\right)$ .

(c) $\mathop{\mathrm{ph}\/}\nolimits\mathop{\Psi_{{3}}\/}\nolimits\!\left(x,y,-3\right)$ .	(d) $\mathop{\mathrm{ph}\/}\nolimits\mathop{\Psi_{{3}}\/}\nolimits\!\left(x,y,-7.5\right)$ .

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

Symbols:: $\mathop{\Psi_{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : real parameter and : real parameter
Keywords:: swallowtail canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F14
Encodings:: pdf, pdf, pdf, pdf, png, png, png, png


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

Figure 36.3.15: Phase of elliptic umbilic canonical integral $\mathop{\mathrm{ph}\/}\nolimits\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!% \left(x,y,0\right)$ .

Symbols:: $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : real parameter and : real parameter
Keywords:: elliptic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F15
Encodings:: pdf, pdf, png, png


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

Figure 36.3.16: Phase of elliptic umbilic canonical integral $\mathop{\mathrm{ph}\/}\nolimits\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!% \left(x,y,2\right)$ .

Symbols:: $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : real parameter and : real parameter
Keywords:: elliptic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F16
Encodings:: pdf, pdf, png, png


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

Figure 36.3.17: Phase of elliptic umbilic canonical integral $\mathop{\mathrm{ph}\/}\nolimits\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!% \left(x,y,4\right)$ .

Symbols:: $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : real parameter and : real parameter
Keywords:: elliptic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F17
Encodings:: pdf, pdf, png, png


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

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

Symbols:: $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : real parameter and : real parameter
Keywords:: hyperbolic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F18
Encodings:: pdf, pdf, png, png


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

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

Symbols:: $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : real parameter and : real parameter
Keywords:: hyperbolic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F19
Encodings:: pdf, pdf, png, png


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

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

Symbols:: $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : real parameter and : real parameter
Keywords:: hyperbolic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F20
Encodings:: pdf, pdf, png, png


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

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

Symbols:: $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : real parameter and : real parameter
Keywords:: hyperbolic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.3.F21
Encodings:: pdf, pdf, png, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 36.2 Catastrophes and Canonical Integrals 36.4 Bifurcation Sets

§36.3 Visualizations of Canonical Integrals

Contents

§36.3(i) Canonical Integrals: Modulus

§36.3(ii) Canonical Integrals: Phase