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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Oblate spheroidal wave function ： ウィキペディア英語版 Oblate spheroidal wave function In applied mathematics, oblate spheroidal wave functions are involved in the solution of the Helmholtz equation in oblate spheroidal coordinates. When solving this equation, $\backslash Delta\; \backslash Phi\; +\; k^2\; \backslash Phi=0$, by the method of separation of variables, $(\backslash xi,\backslash eta,\backslash phi)$, with: :$\backslash \; x=(d/2)\; \backslash xi\; \backslash eta,$ :$\backslash \; y=(d/2)\; \backslash sqrt\; \backslash cos\; \backslash phi,$ :$\backslash \; z=(d/2)\; \backslash sqrt\; \backslash sin\; \backslash phi,$ :$\backslash \; \backslash xi\; \backslash ge\; 0\; \backslash text\; |\backslash eta|\; \backslash le\; 1.$ the solution $\backslash Phi(\backslash xi,\backslash eta,\backslash phi)$ can be written as the product of a radial spheroidal wave function $R\_(-i\; c,i\; \backslash xi)$ and an angular spheroidal wave function $S\_(-i\; c,\backslash eta)$ by $e^$. Here $c=kd/2$, with $d$ being the interfocal length of the elliptical cross section of the oblate spheroid. The radial wave function $R\_(-i\; c,i\; \backslash xi)$ satisfies the linear ordinary differential equation: :$\backslash \; (\backslash xi^2\; +1)\; \backslash frac\; +\; 2\backslash xi\; \backslash frac\; -\backslash left(\backslash lambda\_(c)\; -c^2\; \backslash xi^2\; -\backslash frac\backslash right)\; =\; 0$. The angular wave function satisfies the differential equation: :$\backslash \; (1-\; \backslash eta^2)\; \backslash frac\; -\; 2\backslash eta\; \backslash frac\; +\backslash left(\backslash lambda\_(c)\; +c^2\; \backslash eta^2\; -\backslash frac\backslash right)\; =\; 0$. It is the same differential equation as in the case of the radial wave function. However, the range of the radial coordinate $\backslash xi$ is different from that of the angular coordinate $\backslash eta$. The eigenvalue $\backslash lambda\_(-ic)$ of this Sturm-Liouville differential equation is fixed by the requirement that $$ must be finite for $|\backslash eta|\; =\; 1$. For $c=0$ these two differential equations reduce to the equations satisfied by the associated Legendre polynomials. For $c\backslash ne\; 0$, the angular spheroidal wave functions can be expanded as a series of Legendre functions. The differential equations given above for the oblate radial and angular wave functions can be obtained from the corresponding equations for the prolate spheroidal wave functions by the substitution of $-ic$ for $c$ and $i\; \backslash xi$ for $\backslash xi$. The notation for the oblate spheroidal functions reflects this relationship. There are different normalization schemes for spheroidal functions. A table of the different schemes can be found in Abramowitz and Stegun.〔. M. Abramowitz and I. Stegun. ''Handbook of Mathematical Functions'' (pp. 751-759 ) (Dover, New York, 1972)〕 Abramowitz and Stegun (and the present article) follow the notation of Flammer.〔C. Flammer. ''Spheroidal Wave Functions'' Stanford University Press, Stanford, CA, 1957〕 Originally, the spheroidal wave functions were introduced by C. Niven,〔C. Niven ''(on the conduction of heat in ellipsoids of revolution. )'' Philosophical transactions of the Royal Society of London, 171 p. 117 (1880)〕 which lead to a Helmholtz equation in spheroidal coordinates. Monographs tying together many aspects of the theory of spheroidal wave functions were written by Strutt,〔M. J. O. Strutt. ''Lamesche, Mathieusche and Verdandte Funktionen in Physik und Technik'' Ergebn. Math. u. Grenzeb, 1, pp. 199-323, 1932〕 Stratton et al.,〔J. A. Stratton, P. M. Morse, J. L. Chu, and F. J. Corbato. ''Spheroidal Wave Functions'' Wiley, New York, 1956〕 Meixner and Schafke,〔J. Meixner and F. W. Schafke. ''Mathieusche Funktionen und Sphäroidfunktionen'' Springer-Verlag, Berlin, 1954〕 and Flammer.〔 Flammer〔 provided a thorough discussion of the calculation of the eigenvalues, angular wavefunctions, and radial wavefunctions for both the oblate and the prolate case. Computer programs for this purpose have been developed by many, including Van Buren et al.,〔A. L. Van Buren, R. V. Baier, and S Hanish ''(A Fortran computer program for calculating the oblate spheroidal radial functions of the first and second kind and their first derivatives. )'' (1970)〕 King and Van Buren,〔B. J. King and A. L. Van Buren ''(A Fortran computer program for calculating the prolate and oblate spheroidal angle functions of the first kind and their first and second derivatives. )'' (1970)〕 Baier et al.,〔R. V. Baier, A. L. Van Buren, S. Hanish, B. J. King - (Spheroidal wave functions: their use and evaluation ) The Journal of the Acoustical Society of America, 48, pp. 102–102 (1970)〕 Zhang and Jin,〔S. Zhang and J. Jin. ''Computation of Special Functions'', Wiley, New York, 1996〕 and Thompson.〔W. J. Thomson (Spheroidal Wave functions ) Computing in Science & Engineering p. 84, May–June 1999〕 Van Buren has recently developed new methods for calculating oblate spheroidal wave functions that extend the ability to obtain numerical values to extremely wide parameter ranges. These results are based on earlier work on prolate spheroidal wave functions.〔A. L. Van Buren and J. E. Boisvert. ''Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives'', Quarterly of Applied Mathemathics 60, pp. 589-599, 2002〕〔A. L. Van Buren and J. E. Boisvert. ''Improved calculation of prolate spheroidal radial functions of the second kind and their first derivatives'', Quarterly of Applied Mathematics 62, pp. 493-507, 2004〕 Fortran source code that combines the new results with traditional methods is available at http://www.mathieuandspheroidalwavefunctions.com. Tables of numerical values of oblate spheroidal wave functions are given in Flammer,〔 Hanish et al.,〔S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King ''(Tables of radial spheroidal wave functions, volume 4, oblate, m = 0 )'' (1970)〕〔S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King ''(Tables of radial spheroidal wave functions, volume 5, oblate, m = 1 )'' (1970)〕〔S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King ''(Tables of radial spheroidal wave functions, volume 6, oblate, m = 2 )'' (1970)〕 and Van Buren et al.〔A. L. Van Buren, B. J. King, R. V. Baier, and S. Hanish. ''Tables of Angular Spheroidal Wave Functions, vol. 2, oblate, m = 0'', Naval Research Lab. Publication, U. S. Govt. Printing Office, 1975〕 The Digital Library of Mathematical Functions http://dlmf.nist.gov provided by NIST is an excellent resource for spheroidal wave functions. == References == 〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Oblate spheroidal wave function」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.