= Fundamental strip of a Mellin transform
{title2=$\alpha<\operatorname{Re}s<\beta$}
= Mellin fundamental strip
{c}
{synonym}
The fundamental strip of a <Mellin transform> is the interior of the <set> of <real parts> $\sigma$ for which
$$
\int_0^\infty x^{\sigma-1}|f(x)|\,dx<\infty.
$$
Viewed in the <complex plane>, it is a vertical strip $\alpha<\operatorname{Re}s<\beta$; either bound may be infinite, and the interior may be empty. Both endpoints of the <improper integral> must be checked. Convergence on a boundary line, whether <absolute convergence> or <conditional convergence>, requires a separate test. The <Mellin transform> is <holomorphic> throughout a nonempty fundamental strip.<NIST Digital Library of Mathematical Functions>[Chapter 2, §2.5(i), equation 2.5.1 and the following description of the vertical domain of analyticity]{ref}
A practical sufficient test uses <Big O notation>: if $f(x)=O(x^p)$ near zero and $f(x)=O(x^{-q})$ near infinity, with real $p,q$, then comparison with elementary <improper integrals> gives
$$
\int_0^1x^{\sigma+p-1}\,dx<\infty\ \Longleftrightarrow\ \sigma>-p,\qquad
\int_1^\infty x^{\sigma-q-1}\,dx<\infty\ \Longleftrightarrow\ \sigma<q.
$$
Thus $-p<\sigma<q$ guarantees <absolute convergence>. These upper bounds on $|f|$ need not locate the maximal <Mellin fundamental strip>; faster actual decay can widen it. The strip property itself follows by bounding the weight at any intermediate $\sigma$ by the sum of two weights already known to be integrable. On a closed narrower strip, neighboring integrable weights also dominate the <natural logarithm> factors introduced by <differentiation under the integral sign>, making the transform a <holomorphic function>.
An exact two-endpoint example is
$$
f(x)=\begin{cases}x^2,&0<x\leq1,\\x^{-3},&x>1,\end{cases}\qquad
\mathcal Mf(s)=\frac1{s+2}+\frac1{3-s},\qquad -2<\operatorname{Re}s<3.
$$
Here both bounds are sharp: at $\sigma=-2$ the zero-end <absolute convergence> test contains $\int_0^1dx/x$, and at $\sigma=3$ the infinity-end test contains $\int_1^\infty dx/x$. The displayed <rational function> continues meromorphically outside the <Mellin fundamental strip> and has <simple poles> at $-2$ and $3$. Its other finite continued values outside the strip are values of a <meromorphic continuation>, not of a convergent defining <integral>. More general endpoint information leads to <meromorphic continuation of a Mellin transform from an asymptotic expansion>.
\Image[/mellin-fundamental-strip-and-integral-values.png]
{title=Two endpoint conditions define the Mellin fundamental strip}
{description=Left: the <Mellin fundamental strip> for $x^2$ below one and $x^{-3}$ above one is $-2<\operatorname{Re}s<3$. Crosses mark the two <poles> of the continued <rational function>, not whole lines of <isolated singularities>. Right: the formula and independently computed <integrals> agree on the real slice inside the strip. The curve is drawn only where the defining <integral> converges.}
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