I have an application where it would be very useful to take the Mellin transform of a shifted function. Specifically \begin{equation} M(f(y-x))(y \rightarrow s) = \int_{y=0}^{\infty} f\left(y-x\right) y^{s-1}\, dy. \end{equation} Assuming $x>0$, the substitution $y=\alpha x$, hence $dy = x d\alpha$, leads to \begin{equation} x^{s} M(f((\alpha - 1)x))(\alpha \rightarrow s) = M(f(\alpha - 1))(\alpha \rightarrow s). \end{equation} I have good reason to think this result is wrong, but cannot see the error. Does anyone know if the above has a simple answer?
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