Mehler–Fock transform

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

It is given by

F(x)=\int _{0}^{\infty }P_{it-1/2}(x)f(t)dt,\quad (1\leq x\leq \infty ),

where P is a Legendre function of the first kind.

Under appropriate conditions, the following inversion formula holds:

f(t)=t\tanh(\pi t)\int _{1}^{\infty }P_{it-1/2}(x)F(x)dx,\quad (0\leq t\leq \infty ).

References

Brychkov, Yu.A.; Prudnikov, A.P. (2001), "m/m063340", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
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 (N.S.), 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), Springer Berlin / Heidelberg, 18 (2): 161–194, doi:10.1007/BF01445847, ISSN 0025-5831
Yakubovich, S. B. (2001), "m/m120190", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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