Inversion of the Laplace transform

Let J% exp(-PQf(t) dt = F(P) P > 0 where b > 0 is a given number. We give a formula for f (t) in terms of F (p) , p > 0. The object of this Letter is to give a formula for the inverse Laplace transform in which the data used are given on the semi-axis p > 0 only. Let P>O where b > 0 is a fixed number. We have (1, formula 4.14.1) where Jo is a Bessel function. It follows from (1) and (2) that s, " dt(t2 +x2)-'/Zf(t)=F1(x) x > o where FIW= Som F(PVO(P4 dP. Let t2 =r, x2 =&Fl((c)=F2(<), (2 ~ " ~)-' f (z ' / ~) = f , (r). Let r = b 2 s-' , (= b 2 q-' , f i (b ' ~-') b s-~ / ~ = f 2 (~) , q-1/2F2(b2q-')=F3(q).