e^(ikx)の和

\begin{align*} \sum_{k=-n}^{n}e^{ikx} & =\sum_{k=0}^{2n}e^{i(k-n)x}\\ & =e^{-inx}\sum_{k=0}^{2n}e^{ikx}\\ & =e^{-inx}\frac{1-e^{i(2n+1)x}}{1-e^{ix}}\\ & =\frac{e^{-inx}-e^{i(n+1)x}}{1-e^{ix}}\\ & =\frac{e^{-i\left(n+\frac{1}{2}\right)x}-e^{i\left(n+\frac{1}{2}\right)x}}{e^{-i\frac{x}{2}}-e^{i\frac{x}{2}}}\\ & =\frac{\sin\left\{ \left(n+\frac{1}{2}\right)x\right\} }{\sin\frac{x}{2}} \end{align*}