Accurate power series for eigenvalues of spheroidal angle functions and their convergence radii

Abstract

In this paper, the radius of convergence of the spheroidal power series associated with the eigenvalue is calculated without using the branch point approach. Studying the properties of the power series using the recursion relations among its coefficients in the new method offers some insights into the spheroidal power series and its associated eigenfunction. This study also used the least squares method to accurately compute the convergence radii to five or six significant digits. Within the circle of convergence in the complex plane of the parameter c = kF, where k is the wavenumber and F is the semifocal length of the spheroidal system, the extremely fast convergent spheroidal power series are computed with full precision. In addition, a formula for the magnitude of the upper bound of the error is obtained.Résumé : Nous calculons ici le rayon de convergence de la série de puissance sphéroïdale associée aux valeurs propres sans utiliser l'approche du point de branchement. L'étude des propriétés des séries de puissance par méthode récursive entre les coefficients est une approche qui offre une nouvelle perception des séries de puissance sphéroïdales et des fonctions propres associées. Nous utilisons la méthode des moindres carrés pour calculer de façon précise les rayons de convergence à cinq ou six chiffres significatifs. Les séries de puissance à convergence extrêmement rapide sont calculées avec une complète précision à l'intérieur du rayon de convergence dans le plan complexe du paramètre c = kF, où k est le nombre d'onde et F la demi-longueur focale du système sphéroïdal. Nous obtenons aussi une formule donnant la limite d'erreur supérieure du calcul.[Traduit par la Rédaction]

Accurate Positions of Branch Points of Minimum Magnitudes and Their Associated Spheroidal Eigenvalues

“…INTRODUCTION In this paper, the positions c b of the branch points of minimum magnitudes in the complex plane will be accurately calculated with c=kF, where k is the operating complex valued wavenumber and F is the spheroidal semifocal length, and their associated eigenvalues were also calculated. It follows that the magnitudes of these branch points are the convergence radii |c b | of the spheroidal power series, as recently published in [1].…”

“…The spheroidal angle function of the first kind, , satisfies the following differential equation [1,[9][10][11][12][13][14][15]:…”

“…Research on the branch points has been done by many authors [1][2][3][4][5][6][7][8]. Meixner, Schafke and Wolf [2] may have been the first persons to calculate the convergence radius of the power series of a spheroidal angle function by using a spheroidal branch point of minimum magnitude.…”

“…Moreover, the Newton-Raphson method is chosen even though it is difficult to implement it usually provides a quadratic convergence if the initial trial values are suitably chosen. Moreover, this algorithm relies on the knowledge of the convergence radii whose values are known accurately [1]. Then, the search of these branch points is only in the one dimensional direction (the angular direction in the first quadrant of the complex plane c).…”

“…In the present efficient algorithm to compute these branch points and their associated eigenvalues the complex variables c and its complex eigenvalue λ are solved simultaneously by using the Newton-Raphson method for 2 complex variables cast in matrix form in which both the spheroidal eigenvalue equation and the condition are satisfied. The initial guess of the complex eigenvalue λ and the branch point c is facilitated from the power series given in [1] with about 20 terms and from the approximate formulas of the convergence radii |c b | derived by the author [1]. As long as the equations are exact the algorithm should provide any required accuracy.…”