ABSTRACT
F {g(ξ)}=G(k) = 1√ 2π
e−ikξg(ξ)dξ, (8.2.1)
F −1{G(k)}= g(ξ) = 1√ 2π
eikξG(k)dk. (8.2.2)
Making the changes of variables exp(ξ) = x and ik= c− p, where c is a constant, in results (8.2.1) and (8.2.2) we obtain
G(ip− ic) = 1√ 2π
xp−c−1g(log x)dx, (8.2.3)
g(log x) = 1√ 2π
xc−pG(ip− ic)dp. (8.2.4)
We now write 1√ 2π
x−cg(log x)≡ f(x) and G(ip− ic)≡ f˜(p) to define the Mellin transform of f(x) and the inverse Mellin transform as
M {f(x)}= f˜(p) = ∞∫ 0
xp−1f(x)dx, (8.2.5)
M −1{f˜(p)}= f(x) = 1 2πi
x−pf˜(p)dp, (8.2.6)
where f(x) is a real valued function defined on (0,∞) and the Mellin transform variable p is a complex number. Sometimes, the Mellin transform of f(x) is denoted explicitly by f˜(p) =M [f(x), p]. Obviously, M and M −1 are linear integral operators.
