複素フーリエ級数
複素フーリエ級数
\[
\begin{align}
f(x)&\sim\sum_{n=-\infty}^\infty\left(f(t),\frac{e^{i\frac{2\pi}{T}nt}}{\sqrt{T}}\right)\frac{e^{i\frac{2\pi}{T}nx}}{\sqrt{T}}\\
&=\frac{1}{T}\sum_{n=-\infty}^\infty\left(f(t),e^{i\frac{2\pi}{T}nt}\right)e^{i\frac{2\pi}{T}nx}\\
&=\frac{1}{T}\sum_{n=-\infty}^\infty\left(\int_{-\frac{T}{2}}^\frac{T}{2}f(t)e^{-i\frac{2\pi}{T}nt}dt\right)e^{i\frac{2\pi}{T}nx}
\end{align}
\]
特に、\(T=2\pi\) のとき
\[
f(x)\sim\frac{1}{2\pi}\sum_{n=-\infty}^\infty\left(\int_{-\pi}^\pi f(t)e^{-int}dt\right)e^{inx}
\]