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

[math]\displaystyle{ F(x) =\int_0^\infty P_{it-1/2}(x)f(t) dt,\quad (1 \leq x \leq \infty), }[/math]

where P is a Legendre function of the first kind.

Under appropriate conditions, the following inversion formula holds:

[math]\displaystyle{ f(t) = t \tanh(\pi t) \int_1^\infty P_{it-1/2}(x)F(x) dx ,\quad (0 \leq t \leq \infty). }[/math]

References

• Hazewinkel, Michiel, ed. (2001), "m/m063340", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=m/m063340 
• 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 
• Mehler, F. G. (1881), "Ueber eine mit den Kugel- und Cylinderfunctionen verwandte Function und ihre Anwendung in der Theorie der Elektricitätsvertheilung" (in German), Mathematische Annalen (Springer Berlin / Heidelberg) 18 (2): 161–194, doi:10.1007/BF01445847, ISSN 0025-5831 
• Hazewinkel, Michiel, ed. (2001), "m/m120190", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=m/m120190