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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 $K$

… ►Special cases: $K=1$, fold catastrophe; $K=2$, cusp catastrophe; $K=3$, swallowtail catastrophe. … ►

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: 36.4 Bifurcation Sets

§36.4 Bifurcation Sets

… ► $K=3$, swallowtail bifurcation set: … ►Swallowtail self-intersection line: … ►Swallowtail cusp lines (ribs): … ►

§36.4(ii) Visualizations

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3: 36.5 Stokes Sets

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§36.5(ii) Cuspoids

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$K=3$. Swallowtail

… ►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … ►This consists of a cusp-edged sheet connected to the cusp-edged sheet of the bifurcation set and intersecting the smooth sheet of the bifurcation set. … ►

§36.5(iv) Visualizations

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4: 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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5: 36.7 Zeros

… ►Deep inside the bifurcation set, that is, inside the three-cusped astroid (36.4.10) and close to the part of the $z$-axis that is far from the origin, the zero contours form an array of rings close to the planes …The rings are almost circular (radii close to $(\Delta x)/9$ and varying by less than 1%), and almost flat (deviating from the planes $z_n$ by at most $(\Delta z)/36$). …Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral. … ►

§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 ${\mathrm{\Phi }}^{(\mathrm{H})}$.

6: 20 Theta Functions

Chapter 20 Theta Functions

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7: 36.6 Scaling Relations

§36.6 Scaling Relations

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$\text{umbilics: }{𝐲}^{(\mathrm{U})}(k)=(x{k}^{2/3},y{k}^{2/3},z{k}^{1/3}).$

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Table 36.6.1: Special cases of scaling exponents for cuspoids.

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singularity	$K$	${\beta }_K$	${\gamma }_{1K}$	${\gamma }_{2K}$	${\gamma }_{3K}$	${\gamma }_K$
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swallowtail	3	$\frac{3}{10}$	$\frac{4}{5}$	$\frac{3}{5}$	$\frac{2}{5}$	$\frac{9}{5}$

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8: 36.11 Leading-Order Asymptotics

§36.11 Leading-Order Asymptotics

… ►and far from the bifurcation set, the cuspoid canonical integrals are approximated by … ►

36.11.4

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\exp z$: exponential function, $o\left(x\right)$: order less than and $x$: real parameter

Referenced by:

§36.11

Permalink:

http://dlmf.nist.gov/36.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

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36.11.5 $${\mathrm{\Psi }}_3\left(0,y,0\right)=\overline{{\mathrm{\Psi }}_3(0,-y,0)}=\exp \left(\frac{1}{4}\mathrm{i}\pi \right)\sqrt{\pi /y}\left(1-(\mathrm{i}/\sqrt{3})\exp \left(\frac{3}{2}\mathrm{i}{(2y/5)}^{5/3}\right)+o\left(1\right)\right),$$ $y\to +\mathrm{\infty }$.

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $y$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

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36.11.6

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $o\left(x\right)$: order less than and $z$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

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9: Bibliography N

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

Encodings:

BibTeX

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A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

Encodings:

BibTeX

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Dislocation lines in the swallowtail diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 463, pp. 343–355.

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External Links:

ISSN 1364-5021, Document, ZentralBlatt

Referenced by:

§36.7(iv).

Encodings:

BibTeX

10: 8 Incomplete Gamma and Related
Functions

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