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Lommel function

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The Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation:

Solutions are given by the Lommel functions sμ,ν(z) and Sμ,ν(z), introduced by Eugen von Lommel (1880),

where Jν(z) is a Bessel function of the first kind and Yν(z) a Bessel function of the second kind.

The s function can also be written as[1]

where pFq is a generalized hypergeometric function.

See also

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References

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  1. ^ Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10)
  • Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1953), Higher transcendental functions. Vol II (PDF), McGraw-Hill Book Company, Inc., New York-Toronto-London, MR 0058756
  • Lommel, E. (1875), "Ueber eine mit den Bessel'schen Functionen verwandte Function", Math. Ann., 9 (3): 425–444, doi:10.1007/BF01443342
  • Lommel, E. (1880), "Zur Theorie der Bessel'schen Funktionen IV", Math. Ann., 16 (2): 183–208, doi:10.1007/BF01446386
  • Paris, R. B. (2010), "Lommel function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
  • Solomentsev, E.D. (2001) [1994], "Lommel function", Encyclopedia of Mathematics, EMS Press
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