Fourier coefficients of a periodic logarithmic singularity
= Fourier coefficients of a periodic logarithmic singularity
{c}
{title2=$c_m\sim-(-1)^m/(2m)$}
For the period-one continuation of $\log|x-1/2|$, symmetry and <integration by parts> give $c_m=-(-1)^m\operatorname{Si}(\pi m)/(\pi m)$ for $m\ne0$. Thus the <sine integral> asymptotic gives $c_m=-(-1)^m/(2m)+1/(\pi^2m^2)+O(m^{-4})$ as $m\to+\infty$. Translation of the logarithmic singularity causes the alternating phase; the endpoint derivative mismatch causes the next term.