Abstract
Spheroidal harmonics and modified Bessel functions have wide applications in scientific and engineering computing. Recursive methods are developed to compute the logarithmic derivatives, ratios, and products of the prolate spheroidal harmonics (\(P_n^m(x)\), \(Q_n^m(x)\), \(n\ge m\ge 0\), \(x>1\)), the oblate spheroidal harmonics (\(P_n^m(ix)\), \(Q_n^m(ix)\), \(n\ge m\ge 0\), \(x>0\)), and the modified Bessel functions (\(I_n(x)\), \(K_n(x)\), \(n\ge 0\), \(x>0\)) in order to avoid direct evaluation of these functions that may easily cause overflow/underflow for high degree/order and for extreme argument. Stability analysis shows the proposed recursive methods are stable for realistic degree/order and argument values. Physical examples in electrostatics are given to validate the recursive methods.
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This work was supported by the National Natural Science Foundation of China [Grant Number 11471281].
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Xue, C., Deng, S. Recursive Computation of Logarithmic Derivatives, Ratios, and Products of Spheroidal Harmonics and Modified Bessel Functions and Applications. J Sci Comput 75, 128–156 (2018). https://doi.org/10.1007/s10915-017-0527-3
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DOI: https://doi.org/10.1007/s10915-017-0527-3
Keywords
- Prolate spheroidal harmonics
- Oblate spheroidal harmonics
- Modified Bessel functions
- Associated Legendre functions
- Continued fraction