= Matched expansion with square-root and logarithmic corrections
{title2=$\epsilon^{1/2},\ \epsilon\log\epsilon$}
A decaying algebraic tail in an inner <asymptotic expansion> can become a fractional-order outer correction after rescaling. For an inner tail proportional to $r^{-1/2}$ and outer variable $x=\epsilon r$, the outer correction is proportional to $\epsilon^{1/2}x^{-1/2}$. If the next outer correction contains $\epsilon\log x$, expressing it in the inner variable produces both $\epsilon\log r$ and $\epsilon\log\epsilon$. Omitting the latter loses a term larger than a plain $O(\epsilon)$ correction. Matching must include all these scales and constants, rather than only the leading algebraic powers.
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