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Mehler–Fock transform

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In mathematics, the Mehler–Fock transform is an integral transform introduced by Mehler (1881) and rediscovered by Fock (1943).

It is given by

where P is a Legendre function of the first kind.

Under appropriate conditions, the following inversion formula holds:

References

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  • Brychkov, Yu.A.; Prudnikov, A.P. (2001) [1994], "Mehler–Fock transform", Encyclopedia of Mathematics, EMS Press
  • Fock, V. A. (1943), "On the representation of an arbitrary function by an integral involving Legendre's functions with a complex index", C. R. (Doklady) Acad. Sci. URSS, New Series, 39: 253–256, MR 0009665
  • Mehler, F. G. (1881), "Ueber eine mit den Kugel- und Cylinderfunctionen verwandte Function und ihre Anwendung in der Theorie der Elektricitätsvertheilung", Mathematische Annalen (in German), 18 (2), Springer Berlin / Heidelberg: 161–194, doi:10.1007/BF01445847, ISSN 0025-5831
  • Yakubovich, S. B. (2001) [1994], "Mehler–Fock transform", Encyclopedia of Mathematics, EMS Press