# surface harmonics of the first kind

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## 3 matching pages

##### 1: 14.30 Spherical and Spheroidal Harmonics

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14.30.2
$${Y}_{l}^{m}(\theta ,\varphi )=\mathrm{cos}\left(m\varphi \right){\mathsf{P}}_{l}^{m}\left(\mathrm{cos}\theta \right)\text{or}\mathrm{sin}\left(m\varphi \right){\mathsf{P}}_{l}^{m}\left(\mathrm{cos}\theta \right).$$

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${Y}_{l}^{m}(\theta ,\varphi )$ are known as *surface harmonics of the first kind*:*tesseral*for $$ and*sectorial*for $|m|=l$. …##### 2: 10.73 Physical Applications

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►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 $({\nabla}^{2}+{k}^{2})\psi =0$.
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►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.
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►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.
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►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.
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►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 ${g}_{\mathrm{\ell}}={\mathsf{j}}_{\mathrm{\ell}}$, ${\mathsf{y}}_{\mathrm{\ell}}$, ${\mathsf{h}}_{\mathrm{\ell}}^{(1)}$, or ${\mathsf{h}}_{\mathrm{\ell}}^{(2)}$, depending on the boundary conditions.
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##### 3: Bibliography F

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Theta Functions on Riemann Surfaces.
Springer-Verlag, Berlin.
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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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Diffraction of radio waves around the earth’s surface.
Acad. Sci. USSR. J. Phys. 9, pp. 255–266.
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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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Fast computation of incomplete elliptic integral of first kind by half argument transformation.
Numer. Math. 116 (4), pp. 687–719.
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