surface harmonics of the first kind

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1: 14.30 Spherical and Spheroidal Harmonics

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14.30.2 $$Y_l^m(\theta ,\varphi )=\cos \left(m\varphi \right){𝖯}_l^m\left(\cos \theta \right)\text{ or }\sin \left(m\varphi \right){𝖯}_l^m\left(\cos \theta \right).$$

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Defines:

$Y_l^m(\theta ,\varphi )$: surface harmonic of the first kind

Symbols:

: Ferrers function of the first kind, $\cos z$: cosine function, $\sin z$: sine function, $m$: integer and $l$: nonnegative integer

Referenced by:

§14.30(i), Erratum (V1.0.25) for Section 14.30

Permalink:

http://dlmf.nist.gov/14.30.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.30(i), §14.30 and Ch.14

… ► $Y_l^m(\theta ,\varphi )$ are known as surface harmonics of the first kind: tesseral for and sectorial for $|m|=l$. …

2: 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 $({\nabla }^2+k^2)\psi =0$. … ►Consequently, Bessel functions $J_n\left(x\right)$, and modified Bessel functions $I_n\left(x\right)$, are central to the analysis of microwave and optical transmission in waveguides, including coaxial and fiber. … ►Bessel functions enter in the study of the scattering of light and other electromagnetic radiation, not only from cylindrical surfaces but also in the statistical analysis involved in scattering from rough surfaces. … ►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. … ►With the spherical harmonic $Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ defined as in §14.30(i), the solutions are of the form $f=g_{\mathrm{\ell }}(k\rho )Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ with $g_{\mathrm{\ell }}={𝗃}_{\mathrm{\ell }}$, ${𝗒}_{\mathrm{\ell }}$, ${𝗁}_{\mathrm{\ell }}^{(1)}$, or ${𝗁}_{\mathrm{\ell }}^{(2)}$, depending on the boundary conditions. …

3: Bibliography F

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Theta Functions on Riemann Surfaces. Springer-Verlag, Berlin.

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Notes:

Lecture Notes in Mathematics, Vol. 352

External Links:

MathReview (H. M. Farkas), ZentralBlatt

Referenced by:

§21.1, §21.7(ii), Ch.21.

Encodings:

BibTeX

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Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:

Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.

Referenced by:

§19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).

Encodings:

BibTeX

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Diffraction of radio waves around the earth’s surface. Acad. Sci. USSR. J. Phys. 9, pp. 255–266.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§9.1, Notations U, Notations V.

Encodings:

BibTeX

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Travel time surface of a transverse cusp caustic produced by reflection of acoustical transients from a curved metal surface. J. Acoust. Soc. Amer. 95 (2), pp. 650–660.

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External Links:

Document

Referenced by:

§36.14(iv).

Encodings:

BibTeX

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Fast computation of incomplete elliptic integral of first kind by half argument transformation. Numer. Math. 116 (4), pp. 687–719.

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External Links:

ISSN 0029-599X, Document, FullText, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

§19.36(iv), §19.36(iv).

Encodings:

BibTeX

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