= Mellin transform of an interval indicator
{c}
{title2=$\mathcal M\mathbf1_{[b,c]}(s)=(c^s-b^s)/s$}
For $0<b<c<\infty$, the <indicator function> $f(x)=\mathbf1_{[b,c]}(x)$ has <Mellin transform>
$$
\mathcal Mf(s)=\begin{cases}(c^s-b^s)/s,&s\ne0,\\ \log(c/b),&s=0.\end{cases}
$$
The <interval> stays away from both zero and infinity, so the <integral> is <absolutely convergent> for every <complex number> $s$. Its <Mellin fundamental strip> is the whole <complex plane>. Evaluating the <antiderivative> $x^s/s$ proves the formula for $s\ne0$; at $s=0$, integrating $1/x$ gives the real <natural logarithm> $\log(c/b)$. The apparent division by zero is a <removable singularity>, since $c^s-b^s=s(\log c-\log b)+O(s^2)$ by the <Taylor series> of the <exponential function>. The transform is an <entire function>.
For example, $\mathbf1_{[1,4]}$ gives $(4^s-1)/s$ with value $\log4$ at zero. On $s=i\tau$, the <Mellin transform> sums oscillations over the width $\log4$ of the <interval> in <natural logarithm> coordinates; this is the same cancellation mechanism as in the <Fourier transform of an interval indicator>.
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