Past exam of the mathematics course of the University of Cambridge / 2014 / ib / Paper 4 / 17D / iv / Solution

Evaluate the absolutely convergent complex Fourier series at $x=\pi$. The factors $(-1{)}^n{e}^{in\pi }$ equal one, and the $n=0$ term is $1/{\alpha }^2$. Pairing positive and negative integers givesSince $\sin (\pi \alpha )\ne 0$ for nonintegral complex $\alpha$, rearranging provesThe last term means $\pi \cot (\pi \alpha )/(2\alpha )$, so half-integer arguments are included by the finite cotangent value zero.