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Numerical Algorithms

, Volume 68, Issue 3, pp 497–509 | Cite as

Computation of a numerically satisfactory pair of solutions of the differential equation for conical functions of non-negative integer orders

  • T. M. Dunster
  • A. Gil
  • J. Segura
  • N. M. Temme
Original Paper

Abstract

We consider the problem of computing satisfactory pairs of solutions of the differential equation for Legendre functions of non-negative integer order μ and degree \(-\frac 12+i\tau \), where τ is a non-negative real parameter. Solutions of this equation are the conical functions \(\mbox{{P}}^{\mu }_{-\frac 12+i\tau }(x)\) and \({Q}^{\mu }_{-\frac 12+i\tau }(x)\), x>−1. An algorithm for computing a numerically satisfactory pair of solutions is already available when −1 < x < 1 (see Gil et al. SIAM J. Sci. Comput. 31(3):1716–1741, 2009, Comput. Phys. Commun. 183:794–799, 2012). In this paper, we present a stable computational scheme for a real valued numerically satisfactory companion of the function \(\mbox{{P}}^{\mu }_{-\frac 12+i\tau }(x)\) for x>1, the function \(\Re \left \{e^{-i\pi \mu } {{Q}}^{\mu }_{-\frac {1}{2}+i\tau }(x) \right \}\). The proposed algorithm allows the computation of the function on a large parameter domain without requiring the use of extended precision arithmetic.

Keywords

Legendre functions Conical functions Three-term recurrence relations Numerical methods for special functions 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • T. M. Dunster
    • 1
  • A. Gil
    • 2
  • J. Segura
    • 3
  • N. M. Temme
    • 4
  1. 1.Department of Mathematics and StatisticsSan Diego State UniversitySan DiegoUSA
  2. 2.Departamento de Matemática Aplicada y CC. de la Computación, ETSI CaminosUniversidad de CantabriaSantanderSpain
  3. 3.Departamento de Matemáticas, Estadística y ComputaciónUniversidad de CantabriaSantanderSpain
  4. 4.IAAAbcoudeThe Netherlands

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