= Scaling property of the Mellin transform
{title2=$\mathcal M\{f(a\,\cdot)\}(s)=a^{-s}\mathcal Mf(s)$}
For $a>0$, let $g(x)=f(ax)$. Whenever the <Mellin transform> of $f$ is <absolutely convergent>,
$$
\mathcal Mg(s)=a^{-s}\mathcal Mf(s),\qquad a^{-s}=\exp(-s\log a).
$$
The <Mellin fundamental strip> is unchanged: the corresponding <integrals> of <absolute values> differ by the finite positive factor $a^{-\sigma}$. The <change of variables> $y=ax$ gives $x^{s-1}dx=a^{-s}y^{s-1}dy$, proving the rule. For $s=\sigma+i\tau$, the transform's <absolute value> changes by $a^{-\sigma}$, while the factor $e^{-i\tau\log a}$ changes its <phase>. On the line $\sigma=0$, wherever <absolute convergence> holds, the transform's <absolute value> is preserved.
For the <Mellin transform of an interval indicator>, replacing $\mathbf1_{[1,4]}(x)$ by $\mathbf1_{[1,4]}(2x)=\mathbf1_{[1/2,2]}(x)$ gives
$$
\frac{2^s-(1/2)^s}{s}=2^{-s}\frac{4^s-1}{s}.
$$
At the <removable singularity> $s=0$, both sides have value $\log4$. Scaling leaves the width measured by the <natural logarithm> unchanged. Likewise, the <exponential function> $e^{-ax}$ has <Mellin transform> $a^{-s}\Gamma(s)$ on $\operatorname{Re}s>0$.
\Image[/mellin-scaling-and-logarithmic-translation.png]
{title=Scaling a positive variable translates its logarithmic coordinate}
{description=Left: $f(2x)$ moves the support <interval> $[1,4]$ to $[1/2,2]$. Right: $u=\log x$ turns the same change into a <translation> to the left by $\log2$; the <interval> width remains $\log4$. The <scaling property of the Mellin transform> multiplies the transform by $2^{-s}$.}
Back to article page
