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11: 30.13 Wave Equation in Prolate Spheroidal Coordinates

§30.13 Wave Equation in Prolate Spheroidal Coordinates

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§30.13(i) Prolate Spheroidal Coordinates

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§30.13(ii) Metric Coefficients

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§30.13(iii) Laplacian

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12: 10.73 Physical Applications

… ►In cylindrical coordinates $r$, $\varphi$, $z$, (§1.5(ii) we have … ►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … ►On separation of variables into cylindrical coordinates, the Bessel functions $J_n\left(x\right)$, and modified Bessel functions $I_n\left(x\right)$ and $K_n\left(x\right)$, all appear. … ►The functions ${𝗃}_n\left(x\right)$, ${𝗒}_n\left(x\right)$, ${𝗁}_n^{(1)}\left(x\right)$, and ${𝗁}_n^{(2)}\left(x\right)$ arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates $\rho ,\theta ,\varphi$ (§1.5(ii)): …

13: 28.32 Mathematical Applications

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§28.32(i) Elliptical Coordinates and an Integral Relationship

►If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient. … ► ►

§28.32(ii) Paraboloidal Coordinates

►The general paraboloidal coordinate system is linked with Cartesian coordinates via …

14: 14.30 Spherical and Spheroidal Harmonics

… ►As an example, Laplace’s equation ${\nabla }^{2}W=0$ in spherical coordinates (§1.5(ii)): … ►Here, in spherical coordinates, $L^2$ is the squared angular momentum operator: …

15: 8 Incomplete Gamma and Related
Functions

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16: 28 Mathieu Functions and Hill’s Equation

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17: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

18: 23 Weierstrass Elliptic and Modular
Functions

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19: 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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20: 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 …