What is the relationship between Mellin transforms and integrals?

[eljose]

let,s suppose we have the equality:

f(x)g(x)=H(x) now we have that f(x) and H(x) have no Mellin transform..then would be fair to do this?..

$$f(x)=\frac{1}{2i\pi}\int_{c-i\infty}^{c+i\infty}t^{-s}\frac{M[H(x)]}{M[g(x)]}ds$$

 

[eljose]

Another question..does the Mellin transform of $$ln\zeta(as)$$ exist? where a >0 and real

and the transform of some of derivatives of $$ln\zeta(as)$$ ?

 

[eljose]

another question let be the integral:

$$M(s)=\int_0^{\infty}f(t)t^{s-1}$$ then we take the derivative respecto to s

$$dM(s)/ds=\int_0^{\infty}ln(t)f(t)t^{s-1}$$ so then if dM(s)/ds=r(s) then:

$$M(s)=\int_s^{\infty}r(p)dp$$ is that correct?..thanx