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The asymptotic expansion of the oblate spheroidal eigenfunctions can be expanded in terms of the Laguerre functions of the first and second kinds, from which their asymptotic eigenvalue can be expressed in an inverse power series of c, where the parameter c is proportional to the operating frequency. Analytical expressions of the eigenvalue coefficients, as well as those of the expansion coefficients of the eigenfunctions, are derived and verified with numerical results. PACS Nos.: 02.30Gp, 03.65ge

The asymptotic expansions of the Mathieu eigenfunctions and the prolate spheroidal wave functions can be expanded in terms of the parabolic cylinder functions, from which their asymptotic eigenvalue can be expressed in an inverse power series of c, where the parameter c is proportional to the operating wave number. Analytical expressions of the eigenvalues, as well as those of the expansion coefficients of the eigenfunctions, are derived and verified with numerical results. PACS. No.: 2.30 MV Résumé : On peut faire l'expansion des fonctions propres de Mathieu et des fonctions d'onde sphéroïdales allongées (prolate) en terme des fonctions du cylindre parabolique, d'où il est possible d'écrire les valeurs propres asymptotiques en série de puissance inverse de c, où c est proportionnel au nombre d'onde. Nous obtenons des expressions analytiques pour les valeurs propres et pour les coefficients de l'expansion des fonctions propres et nous les vérifions à l'aide d'un calcul numérique. [Traduit par la rédaction]

Abstract.The series expansion of the prolate radial functions of the second kind, expressed in terms of the spherical Neumann functions, converges very slowly when the spheroid's surface coordinate £ approaches 1 (thin spheroids). In this paper an analytical series expansion in powers of (£2 -1) is obtained to facilitate the convergence. Then, by using the Wronskian test, it is shown that this newly developed expansion has been computed with a double precision accuracy.

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