paraboloidal%20coordinates

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11: 14.31 Other Applications

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§14.31(i) Toroidal Functions

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§14.31(ii) Conical Functions

►The conical functions

\mathsf{P}^{m}_{-\frac{1}{2}+i\tau}\left(x\right)

appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). … ►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …

12: 28.27 Addition Theorems

… ►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. …

13: 30.14 Wave Equation in Oblate Spheroidal Coordinates

§30.14 Wave Equation in Oblate Spheroidal Coordinates

►

§30.14(i) Oblate Spheroidal Coordinates

►Oblate spheroidal coordinates

\xi,\eta,\phi

are related to Cartesian coordinates

x, y, z

by … ►

§30.14(ii) Metric Coefficients

… ►

§30.14(iii) Laplacian

…

14: 23.21 Physical Applications

… ►

§23.21(iii) Ellipsoidal Coordinates

►Ellipsoidal coordinates

(\xi,\eta,\zeta)

may be defined as the three roots

\rho

of the equation …where

x, y, z

are the corresponding Cartesian coordinates and

e_{1}

e_{2}

e_{3}

are constants. The Laplacian operator

\nabla^{2}

(§1.5(ii)) is given by ►

23.21.2

(\eta-\zeta)(\zeta-\xi)(\xi-\eta)\nabla^{2}=(\zeta-\eta)f(\xi)f^{\prime}(\xi)% \frac{\partial}{\partial\xi}+(\xi-\zeta)f(\eta)f^{\prime}(\eta)\frac{\partial}% {\partial\eta}+(\eta-\xi)f(\zeta)f^{\prime}(\zeta)\frac{\partial}{\partial% \zeta},

ⓘ

Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\xi$ : ellipsoidal coordinate, $\eta$ : ellipsoidal coordinate, $\zeta$ : ellipsoidal coordinate and $f(\rho)$ : function
Permalink:: http://dlmf.nist.gov/23.21.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

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15: 30.13 Wave Equation in Prolate Spheroidal Coordinates

§30.13 Wave Equation in Prolate Spheroidal Coordinates

►

§30.13(i) Prolate Spheroidal Coordinates

… ►

§30.13(ii) Metric Coefficients

… ►

§30.13(iii) Laplacian

… ►

16: 10.73 Physical Applications

… ►In cylindrical coordinates

r

\phi

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

\mathsf{j}_{n}\left(x\right)

\mathsf{y}_{n}\left(x\right)

{\mathsf{h}^{(1)}_{n}}\left(x\right)

, and

{\mathsf{h}^{(2)}_{n}}\left(x\right)

arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates

\rho,\theta,\phi

(§1.5(ii)): …

17: 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,

\mathrm{L}^{2}

is the squared angular momentum operator: …

18: 8 Incomplete Gamma and Related
Functions

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19: 28 Mathieu Functions and Hill’s Equation

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20: 8.26 Tables

… ►

•

Khamis (1965) tabulates $P\left(a,x\right)$ for $a=0.05(.05)10(.1)20(.25)70$ , $0.0001\leq x\leq 250$ 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 $(x+n)e^{x}E_{n}\left(x\right)$ 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; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

•

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.