Asymptotic expansions of the oblate spheroidal eigenvalues and wave functions for large parameter <i>c</i>

Do-Nhat, T.

doi:10.1139/p01-034

Cited by 7 publications

(9 citation statements)

References 8 publications

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“…The asymptotic expansion typically applied to the oblate ASWF (OASWF) (case 3 of Equation (1)) and its characteristic eigenvalue for the case of purely imaginary c, or c ⇒ −ic, consists of a similar method to that described by Equations (8)- (12) for the PSWF [19,31]. Similar to the case above, we designate all asymptotic expansions typically applied to the OSWF for purely imaginary c as oblate-type although as will be seen in Section 3.3, this expansion can also be applied to the PSWF in the case of arbitrary complex c. The original differential equation for the OASWF is…”

Section: Oblate-type Asymptotic Expansionmentioning

confidence: 99%

“…Miles [22,23] provided compact, Bessel and Airy function expansions (used in part in Section 5) for [24] (for the prolate case), and Cloizeaux and Mehta [25], also made strides in providing uniform asymptotic expansions for Equation (1), and Sink and Eu [26] provided a WKB approximation for the ASWF. More recently, asymptotic expansions of λ mn for large c or n, m = 0 have been proposed in [27], expansions of Equation (1) as they relate to the quantum shell model for both large n and large c have been proposed by [28,29], and the work of Do-Nhat [30,31] summarizes and provides more details of Flammer's expansions for Equation (1) cases 1 and 3. Evaluation of the SWFs for complex size parameter c = c r + c i i, where c r = {c} and c i = {c}, has received less attention for both the standard and asymptotic regimes.…”

Section: Introductionmentioning

confidence: 99%

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On the Asymptotic Expansion of the Spheroidal Wave Function and Its Eigenvalues for Complex Size Parameter

Barrowes¹,

O’Neill²,

Grzegorczyk³

et al. 2004

Stud Appl Math

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We provide a rapid and accurate method for calculating the prolate and oblate spheroidal wave functions (PSWFs and OSWFs), S mn (c, η), and their eigenvalues, λ mn , for arbitrary complex size parameter c in the asymptotic regime of large |c|, m and n fixed. The ability to calculate these SWFs for large and complex size parameters is important for many applications in mathematics, engineering, and physics. For arbitrary arg(c), the PSWFs and their eigenvalues are accurately expressed by established prolate-type or oblate-type asymptotic expansions. However, determining the proper expansion type is dependent upon finding spheroidal branch points, c mn •;r , in the complex c-plane where the PSWF alternates expansion type due to analytic continuation. We implement a numerical search method for tabulating these branch points as a function of spheroidal parameters m, n, and arg(c). The resulting table allows rapid determination of the appropriate asymptotic expansion type of the SWFs. Normalizations, which are dependent on c, are derived for both the prolateand oblate-type asymptotic expansions and for both (n − m) even and odd. The ordering for these expansions is different from the original ordering of the SWFs and is dictated by the location of c mn •;r . We document this ordering for the specific case of arg(c) = π/4, which occurs for the diffusion equation in spheroidal coordinates. Some representative values of λ mn and S mn (c, η) for large, complex c are also given.

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Section: Oblate-type Asymptotic Expansionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the Asymptotic Expansion of the Spheroidal Wave Function and Its Eigenvalues for Complex Size Parameter

Barrowes¹,

O’Neill²,

Grzegorczyk³

et al. 2004

Stud Appl Math

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“…Other asymptotic results based on WKB methods have been obtained by Sink, and Eu [13]. Recently, asymptotic expansions of λ m ℓ (c) for large l, and c have been proposed by Guimaräes [14], de Moraes and Guimaräes [15], and the work of Do-Nhat [16,17] summarizes and provides more details of Flammer's expansions for λ m ℓ (c). Nevertheless, because of the complexity of the angular spheroidal wave equation with arbitrary complex size parameter c = c r + c i i, where c r = Re{c}, and c i = Im{c}, evaluation of λ m ℓ (c) in this regime has been much less studied.…”

Section: Introductionmentioning

confidence: 99%

The asymptotic iteration method for the angular spheroidal eigenvalues with arbitrary complex size parameter c

Barakat¹,

Abodayeh²,

Abdallah³

et al. 2006

Can. J. Phys.

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“…In the asymptotic regime in which the parameter c = kF (k, operating wavenumber; F, semifocal length of a spheroid) is much greater than one, research was pioneered by Flammer [1] and was subsequently developed by Do-Nhat [2,3] for prolate and oblate spheroidal eigenvalues. In this regime, the work of Barrowes et al [4] was extended to c being a complex variable to compute the branch points of the spheroidal eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

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

Do-NhatTam¹

2011

Can. J. Phys.

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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]

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Asymptotic expansions of the oblate spheroidal eigenvalues and wave functions for large parameter c

Cited by 7 publications

References 8 publications

On the Asymptotic Expansion of the Spheroidal Wave Function and Its Eigenvalues for Complex Size Parameter

On the Asymptotic Expansion of the Spheroidal Wave Function and Its Eigenvalues for Complex Size Parameter

The asymptotic iteration method for the angular spheroidal eigenvalues with arbitrary complex size parameter c

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

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