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


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.


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


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bateman, H.: Higher Transcendental Functions, vol. I. McGraw-Hill (1953)Google Scholar
  2. 2.
    Dunster, T.M.: Conical functions with one or both parameters large. Proc. Roy. Soc. Edinburgh Sect. A 119(3-4), 311–327 (1991)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Dunster, T.M.: Legendre and related functions. In: NIST Handbook of Mathematical Functions, pp. 351–381. Cambridge University Press, New York (2010)Google Scholar
  4. 4.
    Gil, A., Segura, J., Temme, N.M.: Numerical Methods for Special Functions. Society for Industrial and Applied Mathematics (SIAM). Philadelphia (2007)Google Scholar
  5. 5.
    Gil, A., Segura, J., Temme, N.M.: Computing the conical function \(P^{\mu }_{-1/2+i\tau }(x)\). SIAM J. Sci. Comput. 31(3), 1716–1741 (2009)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Gil, A., Segura, J., Temme, N.M.: An improved algorithm and a fortran 90 module for computing the conical function \({P}^{m}_{-1/2+i\tau }(x)\). Comput. Phys. Commun. 183, 794–799 (2012)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Temme, N.M.: The numerical computation of the confluent hypergeometric function U(a,b,z). Numer. Math. 41, 63–82 (1983)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Temme, N.M.: Special Functions: An introduction to the classical functions of mathematical physics. A Wiley-Interscience Publication. Wiley, New York (1996)Google Scholar
  9. 9.
    Thebault, E.: Revised cap harmonic analysis (r-scha): validation and properties. J. Geophys. Res. 111 (2006)Google Scholar

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

Personalised recommendations