Logarithmic singularity ($a\log (z-z_0)$)

A logarithmic singularity behaves locally like $a\log (z-z_0)$ with $a\ne 0$, on a chosen branch. Although unbounded at the endpoint, it is integrable along a finite ray. In a rapidly oscillating endpoint integral its contour rotation produces a logarithm of the large parameter and a branch-angle contribution; this differs from the ordinary smooth-amplitude Watson lemma.