Linear Combination of Mellin Transforms

Theorem

Let $\MM$ be the Mellin transform.

Let $\map f t$, $g \left({t}\right)$ be functions such that $\MM \left\{ {\map f t}\right\} \left({s}\right)$ and $\MM \left\{ {\map f t}\right\} \left({s}\right)$ exist.

Let $\lambda \in \C$ be a constant.

Then:

$\map {\MM \set {\lambda \map f t + \map g t} } s = \lambda \map {\MM \set {\map f t} } s + \map {\MM \set {\map g t} } s$

everywhere all the above expressions are defined.

Proof

\(\ds \map {\MM \set {\lambda \map f t + \map g t} } s\)

\(=\)

\(\ds \int_0^{\to +\infty} t^{s - 1} \paren {\lambda \map f t + \map g t} \rd t\)

Definition of Mellin Transform

\(\ds \)

\(=\)

\(\ds \lambda \int_0^{\to +\infty} t^{s - 1} \map f t \rd t + \int_0^{\to +\infty} t^{s - 1} \map g t \rd t\)

distributing $t^{s - 1}$, Linear Combination of Complex Integrals

\(\ds \)

\(=\)

\(\ds \lambda \map {\MM \set {\map f t} } s + \map {\MM \set {\map g t} } s\)

Definition of Mellin Transform

$\blacksquare$