= Logarithmic endpoint Fourier asymptotics
{title2=$\int_0^1\log t\,e^{ixt}dt\sim[-\pi/2-i(\log x+\gamma)]/x-e^{ix}\sum_{k\ge0}k!/(ix)^{k+2}$}
For real $x\to+\infty$, deformation into the upper half-plane separates a logarithmically singular endpoint at zero from the smooth endpoint at one. The first contributes the exact logarithmic term displayed. The second yields its factorial <asymptotic expansion> by <Watson's lemma>. The two endpoints must both be retained; repeated <integration by parts> without treating the logarithmic singularity loses its leading contribution.
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