Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives


Authors: Arnie L. Van Buren and Jeffrey E. Boisvert
Journal: Quart. Appl. Math. 60 (2002), 589-599
MSC: Primary 65D20; Secondary 33E12
DOI: https://doi.org/10.1090/qam/1914443
MathSciNet review: MR1914443
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Abstract: Alternative expressions for calculating the prolate spheroidal radial functions of the first kind $ R_{ml}^{\left( 1 \right)}\left( c, \xi \right)$ and their first derivatives with respect to $ \xi $ are shown to provide accurate values, even for low values of $ l - m$ where the traditional expressions provide increasingly inaccurate results as the size parameter $ c$ increases to large values. These expressions also converge in fewer terms than the traditional ones. They are obtained from the expansion of the product of $ R_{ml}^{\left( 1 \right)}\left( c, \xi \right)$ and the prolate spheroidal angular function of the first kind $ S_{ml}^{\left( 1 \right)}\left( c, \eta \right)$ in a series of products of the corresponding spherical functions. King and Van Buren [12] had used this expansion previously in the derivation of a general addition theorem for spheroidal wave functions. The improvement in accuracy and convergence using the alternative expressions is quantified and discussed. Also, a method is described that avoids computer overflow and underflow problems in calculating $ R_{ml}^{\left( 1 \right)}\left( c, \xi \right)$ and its first derivative.


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DOI: https://doi.org/10.1090/qam/1914443
Article copyright: © Copyright 2002 American Mathematical Society


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