Fourier 分析

Parseval

\(\zeta(2)\)

考虑对 \(f(x) = x\) 做 Fourier:

\[ c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} x \cdot \mathrm{e}^{-i n x} \mathrm{d} x = \frac{(-1)^n}{n} i \quad \forall\ n\neq 0 \]

显然有\(c_0 = 0\), 因此根据 Parseval:

\[ \sum_{n=\infty}^{\infty} |c_n|^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} x^2 \mathrm{d} x = \frac{\pi^2}{3} \]

因此

\[ \zeta(2) = \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \]