 Logarithmic endpoint Fourier asymptotics

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.