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21: 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 …

22: 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: …

23: 8 Incomplete Gamma and Related
Functions

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24: 28 Mathieu Functions and Hill’s Equation

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25: 1.9 Calculus of a Complex Variable

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Polar Representation

… ►or in polar form (1.9.3) $u$ and $v$ satisfy …

26: 3.5 Quadrature

… ►The steepest descent path is given by $\mathrm{\Im }\left(t-2\sqrt{t}\right)=0$, or in polar coordinates $t=r{\mathrm{e}}^{\mathrm{i}\theta }$ we have $r={\sec }^2\left(\frac{1}{2}\theta \right)$. …

27: 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.

28: 23 Weierstrass Elliptic and Modular
Functions

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29: 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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30: 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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