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

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

where P is a Legendre function of the first kind.

Under appropriate conditions, the following inversion formula holds:

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