のこぎり波\(f\left(x\right)\)のフーリエ係数\(C_{k}\)は
\(k\ne0\)のとき、
\begin{align*} C_{k} & =\frac{1}{2\pi}\int_{-\pi}^{\pi}xe^{-ikx}dx\\ & =\frac{1}{2\pi}\left(\left[\frac{-1}{ik}xe^{-ikx}\right]_{-\pi}^{\pi}+\int_{-\pi}^{\pi}e^{-ikx}dx\right)\\ & =\frac{1}{2\pi}\left(\frac{-1}{ik}\left(\pi e^{-ik\pi}+\pi e^{ik\pi}\right)-\left[\frac{1}{ik}e^{-ikx}\right]_{\pi}^{\pi}\right)\\ & =\frac{1}{2\pi}\left(\frac{-\pi}{ik}\left(e^{-ik\pi}+e^{ik\pi}\right)-\frac{1}{ik}\left(e^{-ik\pi}-e^{ik\pi}\right)\right)\\ & =\frac{1}{2\pi}\left(\frac{-2\pi}{ik}\cos\left(k\pi\right)+\frac{2i}{ik}\sin\left(k\pi\right)\right)\\ & =\frac{i}{k}\cos\left(k\pi\right)\\ & =\frac{\left(-1\right)^{k}i}{k} \end{align*} \(k=0\)のとき、
\begin{align*} C_{0} & =\frac{1}{2\pi}\int_{-\pi}^{\pi}xe^{-i0x}dx\\ & =\frac{1}{2\pi}\int_{-\pi}^{\pi}xdx\\ & =\frac{1}{2\pi}\left[\frac{1}{2}x^{2}\right]_{-\pi}^{\pi}\\ & =\frac{1}{2\pi}\cdot\frac{1}{2}\left(\pi^{2}-\left(-\pi\right)^{2}\right)\\ & =0 \end{align*} となるのでフーリエ級数展開\(F\left(x\right)\)は、
\begin{align*} F\left(x\right) & =\sum_{k=-\infty}^{\infty}C_{k}e^{ikx}\\ & =\sum_{k\ne0}C_{k}e^{ikx}\\ & =\sum_{k=1}^{\infty}\left(C_{k}e^{ikx}+C_{-k}e^{-ikx}\right)\\ & =\sum_{k=1}^{\infty}\left(\frac{\left(-1\right)^{k}i}{k}e^{ikx}-\frac{\left(-1\right)^{k}i}{k}e^{-ikx}\right)\\ & =\sum_{k=1}^{\infty}\frac{\left(-1\right)^{k}i}{k}\left(e^{ikx}-e^{-ikx}\right)\\ & =\sum_{k=1}^{\infty}\frac{\left(-1\right)^{k}i}{k}\cdot2i\sin\left(kx\right)\\ & =2\sum_{k=1}^{\infty}\frac{\left(-1\right)^{k-1}}{k}\sin\left(kx\right) \end{align*} となる。