toroidal coordinates

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11: 32.6 Hamiltonian Structure

… ► ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ can be written as a Hamiltonian system … ►

32.6.3 $$q^{\prime }=p,$$

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Symbols:

$q$: coordinate and $p$: momentum

Referenced by:

§32.6(ii)

Permalink:

http://dlmf.nist.gov/32.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(ii), §32.6 and Ch.32

►

32.6.4 $$p^{\prime }=6q^2+z.$$

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Symbols:

$z$: real, $q$: coordinate and $p$: momentum

Referenced by:

§32.6(ii)

Permalink:

http://dlmf.nist.gov/32.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(ii), §32.6 and Ch.32

… ►

32.6.5 $$\sigma ={\mathrm{H}}_{\text{I}}(q,p,z),$$

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Symbols:

$z$: real, $q$: coordinate, $p$: momentum, $\mathrm{H}(q,p,z)$: Hamiltonian and $\sigma$: function

Permalink:

http://dlmf.nist.gov/32.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(ii), §32.6 and Ch.32

… ►

32.6.7 $$q=-{\sigma }^{\prime },$$

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Symbols:

$q$: coordinate and $\sigma$: function

Permalink:

http://dlmf.nist.gov/32.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(ii), §32.6 and Ch.32

…

12: 36.5 Stokes Sets

… ►For $z\ne 0$, the Stokes set is expressed in terms of scaled coordinates … ►

36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

… ►

36.5.10 $$160u^6+40u^4=Y^2.$$

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Symbols:

$Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

… ►With coordinates … ►

36.5.17 $$Y_{\mathrm{S}}(X)=Y(u,|X|),$$

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Symbols:

$X$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

…

13: 31.17 Physical Applications

… ►Introduce elliptic coordinates $z_1$ and $z_2$ on $S_2$. Then ►

31.17.2 $$\frac{x_s^2}{z_k}+\frac{x_t^2}{z_k-1}+\frac{x_u^2}{z_k-a}=0,$$ $k=1,2$,

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Symbols:

$z$: complex variable, $a$: complex parameter and $x_s$, $x_t$, $x_u$: coordinates

Permalink:

http://dlmf.nist.gov/31.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.17(i), §31.17 and Ch.31

…

14: Howard S. Cohl

… ►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. …

15: 1.5 Calculus of Two or More Variables

… ►

§1.5(ii) Coordinate Systems

… ►

Polar Coordinates

… ►

Cylindrical Coordinates

… ►

Spherical Coordinates

… ►For applications and other coordinate systems see §§12.17, 14.19(i), 14.30(iv), 28.32, 29.18, 30.13, 30.14. …

16: 14.34 Software

… ►

§14.34(iv) Conical (Mehler) and/or Toroidal Functions

…

17: 19.26 Addition Theorems

… ►

19.26.3 $$\sqrt{z}=\frac{\xi {\zeta }^{\prime }+{\eta }^{\prime }\zeta -\xi {\eta }^{\prime }}{\sqrt{\xi \eta {\zeta }^{\prime }}+\sqrt{{\xi }^{\prime }{\eta }^{\prime }\zeta }},$$

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Symbols:

$z$: positive, $\xi$: coordinate, $\eta$: coordinate and $\zeta$: coordinate

Permalink:

http://dlmf.nist.gov/19.26.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(i), §19.26 and Ch.19

►where … ►

19.26.14 $$(p-y)R_C(x,p)+(q-y)R_C(x,q)=(\eta -\xi )R_C(\xi ,\eta ),$$ $x\ge 0$, $y\ge 0$; $p,q\in ℝ\setminus \{0\}$,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\in$: element of, $ℝ$: real line, $\setminus$: set subtraction, $x$: positive, $y$: positive, $\xi$: coordinate and $\eta$: coordinate

Referenced by:

§19.26(i)

Permalink:

http://dlmf.nist.gov/19.26.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(i), §19.26 and Ch.19

… ►

19.26.24 $$z={(\xi \zeta +\eta \zeta -\xi \eta )}^2/(4\xi \eta \zeta ),$$ $(\xi ,\eta ,\zeta )=(x+\lambda ,y+\lambda ,z+\lambda )$,

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Symbols:

$\lambda$: positive, $x$: positive, $y$: positive, $z$: positive, $\xi$: coordinate, $\eta$: coordinate and $\zeta$: coordinate

Permalink:

http://dlmf.nist.gov/19.26.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(iii), §19.26 and Ch.19

…

18: 30.1 Special Notation

… ►

19: Bibliography G

… ►

Evaluation of toroidal harmonics. Comput. Phys. Comm. 124 (1), pp. 104–122.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

►

DTORH3 2.0: A new version of a computer program for the evaluation of toroidal harmonics. Comput. Phys. Comm. 139 (2), pp. 186–191.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

Computing toroidal functions for wide ranges of the parameters. J. Comput. Phys. 161 (1), pp. 204–217.

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External Links:

ISSN 0021-9991, Document, MathReview (Adam Marlewski), ZentralBlatt

Referenced by:

§14.14, §14.15(i), §14.26, §14.31(i), 3rd item.

Encodings:

BibTeX

…

20: 14.1 Special Notation

… ► …