Basic Methods for Computing Special Functions

Abstract

This paper gives an overview of methods for the numerical evaluation of special functions, that is, the functions that arise in many problems from mathematical physics, engineering, probability theory, and other applied sciences. We consider in detail a selection of basic methods which are frequently used in the numerical evaluation of special functions: converging and asymptotic series, including Chebyshev expansions, linear recurrence relations, and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. We give examples of recent software for special functions where these methods are used. We mention a list of new publications on computational aspects of special functions available on our website.

Keywords

Numerical evaluation of special functions Chebyshev expansions Quadrature methods Transformation of series Continued fractions Asymptotic analysis 

Mathematics Subject Classification (2000)

65D20 41A60 33C05 33C10 33C15 

Notes

Acknowledgements

We thank the referees for their helpful comments and Dr. Ernst Joachim Weniger for providing us with notes that we used for writing Sect. 4.7.2. We acknowledge financial support from Ministerio de Educación y Ciencia, project MTM2006–09050. NMT acknowledges financial support from Gobierno of Navarra, Res. 07/05/2008.

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada y CC. de la ComputaciónETSI Caminos, Universidad de CantabriaSantanderSpain
  2. 2.Departamento de Matemáticas, Estadística y ComputaciónUniversidad de CantabriaSantanderSpain
  3. 3.CWIAmsterdamThe Netherlands

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