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 ) = ∫ 0 ∞ P i t − 1 / 2 ( x ) f ( t ) d t , ( 1 ≤ x ≤ ∞ ) , {\displaystyle 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 ⁡ ( π t ) ∫ 1 ∞ P i t − 1 / 2 ( x ) F ( x ) d x , ( 0 ≤ t ≤ ∞ ) . {\displaystyle f(t)=t\tanh(\pi t)\int _{1}^{\infty }P_{it-1/2}(x)F(x)dx,\quad (0\leq t\leq \infty ).}

• 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