[mathjax]
目次
パーセバルの等式
パーセバルの等式とは
\(f(x)\)のフーリエ級数の係数に関して、以下がパーセバルの等式です。
\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} {f(x)}^2 dx=\displaystyle\frac{a_{0}^2}{2}+\displaystyle\sum_{n=1}^{\infty}(a_{n}^2+b_{n}^2)\)
ただし
\(a_{0}=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x) dx\)
\(a_{n}=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x) \cos nx dx\)
\(b_{n}=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi}f(x) \sin nx dx\)
証明
\(f(x)=\displaystyle\frac{a_{0}}{2}+\displaystyle\sum_{n=1}^{\infty}(a_{n}\cos nx+b_{n}\sin nx)\)
というフーリエ級数式を使って計算する。
\(左辺=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x)\cdot f(x) dx\)
\(=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x)\cdot \biggl[\displaystyle\frac{a_{0}}{2}+\displaystyle\sum_{n=1}^{\infty}(a_{n}\cos nx+b_{n}\sin nx)\biggr] dx\)
\(=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi}\displaystyle\frac{a_{0}}{2}f(x) dx+\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi}\displaystyle\sum_{n=1}^{\infty}a_{n}f(x)\cos nx dx+\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi}\displaystyle\sum_{n=1}^{\infty}b_{n}f(x)\sin nx dx\)
\(=\displaystyle\frac{a_{0}}{2}\cdot\)\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x) dx\)\(+\displaystyle\sum_{n=1}^{\infty}a_{n}\cdot\)\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x) \cos nx dx\)\(+\displaystyle\sum_{n=1}^{\infty}b_{n}\cdot\)\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x) \sin nx dx\)
\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x) dx=a_{0}\)
\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} f(x) \cos nx dx=a_{n}\)
\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi}f(x) \sin nx dx=b_{n}\)
を代入すると、
\(=\displaystyle\frac{a_{0}^2}{2}+\displaystyle\sum_{n=1}^{\infty}(a_{n}^2+b_{n}^2)=右辺\) となる。
例題
1番
\(\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{1}{n^2}=\displaystyle\frac{\pi^2}{6}\) を示す。
\(f(x)=x\) とおく。奇関数なので、\(a_{0}\)、\(a_{n}\)は0。\(b_{n}\)を部分積分で求めると
\(b_{n}=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} x\sin nx dx=\displaystyle\frac{2}{n}(-1)^{n-1}\)
これらをパーセバルの等式に代入する。
\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi}x^2 dx=\displaystyle\frac{a_{0}^2}{2}+\displaystyle\sum_{n=1}^{\infty}(a_{n}^2+b_{n}^2)\)
\(左辺=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} x^2 dx=\displaystyle\frac{2}{3}\pi^2\)
\(右辺=\displaystyle\sum_{n=1}^{\infty}\biggl[\displaystyle\frac{2}{n}(-1)^{n-1}\biggr]^2=\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{4}{n^2}\)
よって \(\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{1}{n^2}=\displaystyle\frac{\pi^2}{6}\)
2番
\(\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{1}{n^4}=\displaystyle\frac{\pi^4}{90}\) を示す。
\(f(x)=x^2\) とおく。偶関数なので、\(b_{n}\)は0。\(a_{0}\)と\(a_{n}\)をそれぞれ求めると
\(a_{0}=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} x^2 dx=\displaystyle\frac{2}{3}\pi^2\)
\(a_{n}=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} x^2\cos nx dx=\displaystyle\frac{4}{n^2}(-1)^n\)
これらをパーセバルの等式に代入する。
\(\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi}x^4 dx=\displaystyle\frac{a_{0}^2}{2}+\displaystyle\sum_{n=1}^{\infty}(a_{n}^2+b_{n}^2)\)
\(左辺=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\pi}^{\pi} x^4 dx=\displaystyle\frac{2}{5}\pi^4\)
\(右辺=\displaystyle\frac{2}{9}\pi^2+\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{16}{n^4}\)
よって \(\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{1}{n^4}=\displaystyle\frac{1}{16}\biggl(\displaystyle\frac{2}{5}\pi^4-\displaystyle\frac{2}{9}\pi^4\biggr)=\displaystyle\frac{\pi^4}{90}\)
