Translational addition theorems for prolate spheroidal vector wave functions 𝑀^{𝑟} and 𝑁^{𝑟}

Dalmas, Jeannine; Deleuil, Roger

doi:10.1090/qam/856176

Translational addition theorems for prolate spheroidal vector wave functions $\textbf {M}^r$ and $\textbf {N}^r$

Authors: Jeannine Dalmas and Roger Deleuil
Journal: Quart. Appl. Math. 44 (1986), 213-222
MSC: Primary 33A55; Secondary 78A45
DOI: https://doi.org/10.1090/qam/856176
MathSciNet review: 856176
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Abstract: The translational addition theorems for prolate spheroidal vector wave functions ${\textrm {M}^r}$ and ${\textrm {N}^r}$ with reference to the spheroidal coordinate system at the origin 0 are obtained in terms of spheroidal vector wave functions with reference to the translated spheroidal coordinate system at the origin $0’$. These addition theorems are absolutely necessary for the study of the multiple scattering of electromagnetic waves when the single electromagnetic scattering solution is also expressed in terms of spheroidal vector wave functions ${\textrm {M}^r}$ and ${\textrm {N}^r}$.

Bernard Friedman and Joy Russek, Addition theorems for spherical waves, Quart. Appl. Math. 12 (1954), 13–23. MR 60649, DOI https://doi.org/10.1090/S0033-569X-1954-60649-8
Seymour Stein, Addition theorems for spherical wave functions, Quart. Appl. Math. 19 (1961), 15–24. MR 120407, DOI https://doi.org/10.1090/S0033-569X-1961-0120407-5
Orval R. Cruzan, Translational addition theorems for spherical vector wave functions, Quart. Appl. Math. 20 (1962/63), 33–40. MR 132851, DOI https://doi.org/10.1090/S0033-569X-1962-0132851-2

J. Bruning and Y. Lo, Multiple scattering of EM waves by spheres, part I and II, I.E.E.E. Trans. Antennas Prop. 19, 378–400 (1971)

J. Dalmas and R. Deleuil, Diffusion multiple des ondes électromagnétiques par des ellipsoïdes de révolution allongés, Aptica Acta 29, 1117–1131 (1982)

B. P. Sinha and R. H. MacPhie, Electromagnetic plane wave scattering by a system of two parallel conducting prolate spheroids, I.E.E.E. Trans. Antennas Prop. 31, 294–304 (1983)

S. Asano and G. Yamamoto, Light scattering by a spheroidal particle, Appl. Optics 14, 29–49 (1975)

J. Dalmas, Indicatrices de diffusion d’un ellipsoïde de révolution allongé de conduction infinie en incidence oblique, Optica Acta 28, 1277–1287 (1981)

Bateshwar P. Sinha and Robert H. MacPhie, Electromagnetic scattering by prolate spheroids for plane waves with arbitrary polarization and angle of incidence, Radio Sci. 12 (1977), no. 2, 171–184. MR 459329, DOI https://doi.org/10.1029/RS012i002p00171
Bateshwar P. Sinha and Robert H. Macphie, Translational addition theorems for spheroidal scalar and vector wave functions, Quart. Appl. Math. 38 (1980/81), no. 2, 143–158. MR 580875, DOI https://doi.org/10.1090/S0033-569X-1980-0580875-9
Jeannine Dalmas and Roger Deleuil, Établissement d’une formule de récurrence pour l’addition des fonctions d’onde scalaires sphéroïdales intervenant dans l’étude de la diffusion multiple par des ellipsoïdes de révolution, Optica Acta 30 (1983), no. 12, 1697–1705 (French, with English and German summaries). MR 730360, DOI https://doi.org/10.1080/713821116

J. A. Stratton, Electromagnetic theory, McGraw-Hill, 1941

Carson Flammer, Spheroidal wave functions, Stanford University Press, Stanford, California, 1957. MR 0089520
J. Dalmas and R. Deleuil, Diffusion d’une onde électromagnétique par un ellipsoïde de révolution allongé et par un demi-ellipsoïde posé sur un plan en incidence axiale, Optica Acta 27 (1980), no. 5, 637–649 (French, with English summary). MR 577735, DOI https://doi.org/10.1080/713820293

J. Dalmas, Diffusion d’une onde électromagnétique par un ellipsoïde de révolution allongé de conduction infinie en incidence non axiale, Optica Acta 28, 933–948 (1981)

H. Sircar, On the evolution of the definite integrals $\int _{ - 1}^{ + 1} {P_n^m\left ( x \right )P_q^p\left ( x \right )dx}$ and $\int _0^1 {P_n^m\left ( x \right )P_q^p\left ( x \right )dx}$, Proc. Edinburg Math. Soc. 1, 241–245 (1927)

A. R. Edmonds, Angular momentum in quantum mechanics, Investigations in Physics, Vol. 4, Princeton University Press, Princeton, N.J., 1957. MR 0095700