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21: 23 Weierstrass Elliptic and Modular
Functions

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22: 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

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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

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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

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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

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23: 14.19 Toroidal (or Ring) Functions

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§14.19(i) Introduction

… ►This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates $(\eta ,\theta ,\varphi )$, which are related to Cartesian coordinates $(x,y,z)$ by …

24: 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

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25: 36 Integrals with Coalescing Saddles

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26: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

27: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

28: 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. …

29: 33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ for $\eta =0.5(.5)20$ and $\rho =1(1)20$, 5S; $C_0\left(\eta \right)$ for $\eta =0(.05)3$, 6S.

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30: 1.5 Calculus of Two or More Variables

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§1.5(ii) Coordinate Systems

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Polar Coordinates

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Cylindrical Coordinates

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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. …