Mehler–Fock transform explained

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

It is given by

F(x) = \int ^{inf ty} _{0} P _{it-1/2} (x)f(t) dt, ( 1 \leq x \leq inf ty),

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 ^{inf ty} _{1} P _{it-1/2} (x)F(x) dx , ( 0 \leq t \leq inf ty).