canonical denominator (or numerator)

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1: 36.2 Catastrophes and Canonical Integrals

§36.2 Catastrophes and Canonical Integrals

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§36.2(i) Definitions

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Canonical Integrals

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§36.2(ii) Special Cases

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§36.2(iii) Symmetries

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2: 36.3 Visualizations of Canonical Integrals

§36.3 Visualizations of Canonical Integrals

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§36.3(i) Canonical Integrals: Modulus

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§36.3(ii) Canonical Integrals: Phase

►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. …

3: 1.12 Continued Fractions

… ► $A_n$ and $B_n$ are called the $n$th (canonical) numerator and denominator respectively. … ►

1.12.7 $$A_n{B}_{n-1}-B_n{A}_{n-1}={(-1)}^{n-1}\prod _{k=1}^n{a}_k,$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$k$: integer, $n$: nonnegative integer, $A_n$: $n$th numerator and $B_n$: $n$th denominator

Permalink:

http://dlmf.nist.gov/1.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

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1.12.11 $$a_n=\frac{B_n}{B_{n-2}}\frac{C_{n-1}-C_n}{C_{n-1}-C_{n-2}},$$ $n=2,3,4,\mathrm{\dots }$,

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Symbols:

$n$: nonnegative integer, $B_n$: $n$th denominator and $C_n(w)$: continued fraction

Permalink:

http://dlmf.nist.gov/1.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

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1.12.13 $$b_n=\frac{B_n}{B_{n-1}}\frac{C_n-C_{n-2}}{C_{n-1}-C_{n-2}},$$ $n=2,3,4,\mathrm{\dots }$,

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Symbols:

$n$: nonnegative integer, $B_n$: $n$th denominator and $C_n(w)$: continued fraction

Permalink:

http://dlmf.nist.gov/1.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

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4: 36.10 Differential Equations

§36.10 Differential Equations

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§36.10(i) Equations for ${\mathrm{\Psi }}_K\left(𝐱\right)$

►In terms of the normal form (36.2.1) the ${\mathrm{\Psi }}_K\left(𝐱\right)$ satisfy the operator equation … ► $K=3$, swallowtail: … ►In terms of the normal forms (36.2.2) and (36.2.3), the ${\mathrm{\Psi }}^{(\mathrm{U})}\left(𝐱\right)$ satisfy the following operator equations …

5: 36.14 Other Physical Applications

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§36.14(i) Caustics

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§36.14(ii) Optics

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§36.14(iii) Quantum Mechanics

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§36.14(iv) Acoustics

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6: 36.9 Integral Identities

§36.9 Integral Identities

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36.9.1 $${|{\mathrm{\Psi }}_1\left(x\right)|}^2=2^{5/3}{\int }_0^{\mathrm{\infty }}{\mathrm{\Psi }}_1\left(2^{2/3}(3u^2+x)\right)du;$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/36.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

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36.9.8 $${\left|{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,z\right)\right|}^2=8{\pi }^2{\left(\frac{2}{9}\right)}^{1/3}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{Ai}\left({\left(\frac{4}{3}\right)}^{1/3}(x+zv+3u^2)\right)\mathrm{Ai}\left({\left(\frac{4}{3}\right)}^{1/3}(y+zu+3v^2)\right)dudv.$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\int$: integral, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

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36.9.9 $${\left|{\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,z\right)\right|}^2=\frac{8{\pi }^2}{3^{2/3}}{\int }_0^{\mathrm{\infty }}{\int }_0^{2\pi }\mathrm{\Re }\left(\mathrm{Ai}\left(\frac{1}{3^{1/3}}\left(x+\mathrm{i}y+2zu\exp \left(\mathrm{i}\theta \right)+3u^2\exp \left(-2\mathrm{i}\theta \right)\right)\right)\mathrm{Bi}\left(\frac{1}{3^{1/3}}\left(x-\mathrm{i}y+2zu\exp \left(-\mathrm{i}\theta \right)+3u^2\exp \left(2\mathrm{i}\theta \right)\right)\right)\right)udud\theta .$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter and $x$: real parameter

Keywords:

elliptic umbilic canonical integral, integral identity

Permalink:

http://dlmf.nist.gov/36.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

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7: 36.1 Special Notation

§36.1 Special Notation

… ►The main functions covered in this chapter are cuspoid catastrophes ${\mathrm{\Phi }}_K(t;𝐱)$; umbilic catastrophes with codimension three ${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;𝐱)$, ${\mathrm{\Phi }}^{(\mathrm{H})}(s,t;𝐱)$; canonical integrals ${\mathrm{\Psi }}_K\left(𝐱\right)$, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$; diffraction catastrophes ${\mathrm{\Psi }}_K(𝐱;k)$, ${\mathrm{\Psi }}^{(\mathrm{E})}(𝐱;k)$, ${\mathrm{\Psi }}^{(\mathrm{H})}(𝐱;k)$ generated by the catastrophes. …

8: 36.6 Scaling Relations

§36.6 Scaling Relations

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${\mathrm{\Psi }}_K(𝐱;k)=k^{{\beta }_K}{\mathrm{\Psi }}_K\left(𝐲(k)\right),$

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${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)=k^{{\beta }^{(\mathrm{U})}}{\mathrm{\Psi }}^{(\mathrm{U})}\left({𝐲}^{(\mathrm{U})}(k)\right),$

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Indices for $k$-Scaling of Magnitude of ${\mathrm{\Psi }}_K$ or ${\mathrm{\Psi }}^{(\mathrm{U})}$ (Singularity Index)

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9: 36.11 Leading-Order Asymptotics

§36.11 Leading-Order Asymptotics

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

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

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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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36.11.7 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,z\right)=\frac{\pi }{z}\left(\mathrm{i}+\sqrt{3}\exp \left(\frac{4}{27}\mathrm{i}{z}^3\right)+o\left(1\right)\right),$$ $z\to \pm \mathrm{\infty }$,

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Symbols:

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

Permalink:

http://dlmf.nist.gov/36.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

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36.11.8 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,z\right)=\frac{2\pi }{z}\left(1-\frac{\mathrm{i}}{\sqrt{3}}\exp \left(\frac{1}{27}\mathrm{i}{z}^3\right)+o\left(1\right)\right),$$ $z\to \pm \mathrm{\infty }$.

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Symbols:

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

Permalink:

http://dlmf.nist.gov/36.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

10: 36.7 Zeros

§36.7 Zeros

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§36.7(i) Fold Canonical Integral

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§36.7(ii) Cusp Canonical Integral

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§36.7(iii) Elliptic Umbilic Canonical Integral

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