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

$F(x)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{P}_{it-1/2}(x)f(t)dt,(1\le x\le \mathrm{\infty }),$

where P is a Legendre function of the first kind.

Under appropriate conditions, the following inversion formula holds:

$f(t)=t\tanh (\pi t)\underset{1}{\overset{\mathrm{\infty }}{\int }}{P}_{it-1/2}(x)F(x)dx,(0\le t\le \mathrm{\infty }).$

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