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Recent Advances in Computational and Applied Mathematics pp 67–121Cite as

Basic Methods for Computing Special Functions

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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.

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Notes

  1. 1.

    http://mathworld.wolfram.com/.

  2. 2.

    http://en.wikipedia.org/.

  3. 3.

    The “Devil’s invention” refers to a quote from Niels Hendrik Abel (1828), who claimed “Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever.”

  4. 4.

    This terminology is not the same in all branches of applied mathematics and mathematical physics: sometimes one sees a complete interchange of the names ‘Stokes line’ and ‘anti-Stokes line’.

  5. 5.

    http://www.moshier.net.

  6. 6.

    http://www.netlib.org/.

  7. 7.

    http://gams.nist.gov/Classes.html.

  8. 8.

    http://functions.unican.es.

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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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Authors and Affiliations

  1. Departamento de Matemática Aplicada y CC. de la Computación, ETSI Caminos, Universidad de Cantabria, 39005, Santander, Spain

    Amparo Gil

  2. Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, 39005, Santander, Spain

    Javier Segura

  3. CWI, Science Park 123, 1098 XG, Amsterdam, The Netherlands

    Nico M. Temme

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  1. Amparo Gil
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  2. Javier Segura
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  3. Nico M. Temme
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Correspondence to Javier Segura .

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Editors and Affiliations

  1. Dpt. Computer Science and Technology, University of Peloponnese, Tripolis, GR-221 00, Greece

    Theodore E. Simos

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Gil, A., Segura, J., Temme, N.M. (2011). Basic Methods for Computing Special Functions. In: Simos, T. (eds) Recent Advances in Computational and Applied Mathematics. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9981-5_4

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