Laplace equation

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1: 34.12 Physical Applications

§34.12 Physical Applications

… ► $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).

2: 14.31 Other Applications

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§14.31(ii) Conical Functions

►The conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^m\left(x\right)$ 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)). …

3: 29.19 Physical Applications

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§29.19(i) Lamé Functions

►Simply-periodic Lamé functions ($\nu$ noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. …

4: 10.73 Physical Applications

… ►Bessel functions of the first kind, $J_n\left(x\right)$, arise naturally in applications having cylindrical symmetry in which the physics is described either by Laplace’s equation ${\nabla }^{2}V=0$, or by the Helmholtz equation $({\nabla }^2+k^2)\psi =0$. ►Laplace’s equation governs problems in heat conduction, in the distribution of potential in an electrostatic field, and in hydrodynamics in the irrotational motion of an incompressible fluid. … …

5: 19.18 Derivatives and Differential Equations

… ►The next four differential equations apply to the complete case of $R_F$ and $R_G$ in the form $R_{-a}(\frac{1}{2},\frac{1}{2};z_1,z_2)$ (see (19.16.20) and (19.16.23)). … ►and $U=R_{-a}(\frac{1}{2},\frac{1}{2};z+\mathrm{i}\rho ,z-\mathrm{i}\rho )$, with $\rho =\sqrt{x^2+y^2}$, satisfies Laplace’s equation: …

6: 14.30 Spherical and Spheroidal Harmonics

… ►As an example, Laplace’s equation ${\nabla }^{2}W=0$ in spherical coordinates (§1.5(ii)): …

7: 14.19 Toroidal (or Ring) Functions

… ►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 …

8: 18.17 Integrals

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§18.17(vi) Laplace Transforms

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Jacobi

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Laguerre

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Hermite

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18.17.40 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}{L}_n^{(\alpha )}\left(bx\right)x^{z-1}dx=\frac{\mathrm{\Gamma }\left(z+n\right)}{n!}{(a-b)}^n{a}^{-n-z}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,1+\alpha -z}{1-n-z};\frac{a}{a-b}),$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Laplace transform, Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

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9: Bibliography B

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An Introduction to Linear Difference Equations. Dover Publications Inc., New York.

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Notes:

Unaltered republication of the original 1927 Harvard University edition.

External Links:

MathReview, ZentralBlatt

Referenced by:

§8.1, Notations P, Notations Q.

Encodings:

BibTeX

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Some Geometric Aspects of Wave Motion: Wavefront Dislocations, Diffraction Catastrophes, Diffractals. In Geometry of the Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Vol. 36, pp. 13–28.

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External Links:

MathReview (I. Stewart), ZentralBlatt

Referenced by:

§36.14(i).

Encodings:

BibTeX

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From Klein to Painlevé via Fourier, Laplace and Jimbo. Proc. London Math. Soc. (3) 90 (1), pp. 167–208.

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External Links:

ISSN 0024-6115, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.9(iii).

Encodings:

BibTeX

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Laplace approximations for hypergeometric functions with matrix argument. Ann. Statist. 30 (4), pp. 1155–1177.

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External Links:

ISSN 0090-5364, Document, MathReview, ZentralBlatt

Referenced by:

§35.6(iv), §35.7(iv), §35.7(iv), §35.9.

Encodings:

BibTeX

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Laplace approximation for Bessel functions of matrix argument. J. Comput. Appl. Math. 155 (2), pp. 359–382.

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External Links:

ISSN 0377-0427, Document, MathReview (S. Bhargava), ZentralBlatt

Referenced by:

§35.5(ii), §35.5(iii).

Encodings:

BibTeX

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10: 15.4 Special Cases

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15.4.34 $$F(3a,a;2a;{\mathrm{e}}^{\mathrm{i}\pi /3})=\sqrt{\pi }{\mathrm{e}}^{\mathrm{i}\pi a/2}\frac{2^{2a}\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{3^{(3a+1)/2}}\left(\frac{1}{\mathrm{\Gamma }\left(\frac{1}{3}+a\right)\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}+a\right)\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $a$: real or complex parameter

Proof sketch:

From §15.5(ii) one obtains that $w(a)={\mathrm{e}}^{-\mathrm{i}\pi a/2}{2}^{-2a}{3}^{(3a+1)/2}F(3a,a;2a;{\mathrm{e}}^{\mathrm{i}\pi /3})\mathrm{\Gamma }\left(\frac{1}{3}+a\right)/\mathrm{\Gamma }\left(\frac{1}{2}+a\right)$ is a solution of the difference equation $(3a+5)w(a+2)-6(a+1)w(a+1)+(3a+1)w(a)=0$. Two linearly independent solutions of this difference equation are $w_1(a)=1$ and $w_2(a)=\mathrm{\Gamma }\left(\frac{1}{3}+a\right)/\mathrm{\Gamma }\left(\frac{2}{3}+a\right)$. Combining (15.6.1) with Laplace’s method (see §2.4(iii)) we obtain that $w(a)\sim \sqrt{\pi }\left(\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{a^{-1/3}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right)$, as $a\to \mathrm{\infty }$. Hence, $w(a)=\sqrt{\pi }\left(\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{w_2(a)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right)$.

Notes:

See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer.

Referenced by:

§15.4(iii), §15.4(iii), Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/15.4.E34

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.4(iii), §15.4 and Ch.15

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