フーリエ級数の問題

■問題

$3$\displaystyle{ f $x$ = \|x\| }$ $3$\displaystyle{ $ - {\pi}\leq x\leq {\pi} $ }$ ， $3$\displaystyle{f\left({ x+2{\pi} }\right)=f\left({x}\right)}$

のフーリエ級数を求めよ．

■答

$3$\displaystyle{ f $x$ \sim \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{\pi}- \frac{\hspace{2}4\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\left({ \cos x+ \frac{\hspace{2}1\hspace{2}}{\hspace{2}{3}^{2}\hspace{2}}\cos 3x+ \frac{\hspace{2}1\hspace{2}}{\hspace{2}{5}^{2}\hspace{2}}\cos 5x+ \frac{\hspace{2}1\hspace{2}}{\hspace{2}{7}^{2}\hspace{2}}\cos 7x+ \cdots }\right) }$

■ヒント

フーリエ係数

$3$\displaystyle{ \hspace{1}{a}_{0}=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2{\pi} \hspace{2}}\displaystyle{ \int _{ - {\pi} }^{{\pi}} f $x$ dx } }$

$3$\displaystyle{ \hspace{1}{a}_{n}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\int _{ - {\pi} }^{{\pi}} f $x$ \cos nx dx }$ 　　　　( $3$\displaystyle{ n=1,2,3\text{\hspace{1}}\cdot \text{\hspace{1}}\cdot \text{\hspace{1}}\cdot }$ )

$3$\displaystyle{ \hspace{1}{b}_{n}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\int _{ - {\pi} }^{{\pi}} f $x$ \sin nxdx }$ 　　　　( $3$\displaystyle{ n=1,2,3\text{\hspace{1}}\cdot \text{\hspace{1}}\cdot \text{\hspace{1}}\cdot }$ )

を求める．

■解説

$3$\displaystyle{ f $x$ = \|x\| }$ $3$\displaystyle{ $ - {\pi}\leq x\leq {\pi} $ }$

を書き換えると

$3$\displaystyle{ f $x$ = \{ \begin{array} - x & \vspace{6}\\ x& \vspace{6}\\ \end{array}\begin{array} & \vspace{6}\\ & \vspace{6}\\ \end{array}\begin{array} & \vspace{6}\\ & \vspace{6}\\ \end{array} \begin{array} $ - {\pi}\leq x\leq 0 $ & \vspace{6}\\ $ 0\leq x\leq {\pi} $ & \vspace{6}\\ \end{array} }$

となる．

$3$\displaystyle{ \hspace{1}{a}_{0}=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2{\pi} \hspace{2}}\displaystyle{ \int _{ - {\pi} }^{{\pi}} f $x$ dx } }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2{\pi} \hspace{2}} \{ \displaystyle{ \int _{ - {\pi} }^{0} $ - x $ dx+\displaystyle{ \int _{0}^{{\pi}} xdx } } \} }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2{\pi} \hspace{2}} \{ \[ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{x}^{2} \] _{ - {\pi} }^{0}+ \[ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{x}^{2} \] _{0}^{{\pi}} \} }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2{\pi} \hspace{2}} $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{{\pi}}^{2}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{{\pi}}^{2} $ }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2{\pi} \hspace{2}}\cdot {{\pi}}^{2} }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{\pi} }$

$3$\displaystyle{ \hspace{1}{a}_{n}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\displaystyle{ \int _{ - {\pi} }^{{\pi}} f $x$ \cos nxdx } }$

$3$\displaystyle{ f$x$ }$ が偶関数なので、 $3$\displaystyle{ f$x$=\cos x }$ も偶関数である(偶関数の積分を参照)．よって

$3$\displaystyle{ =\frac{\hspace{2}2\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\displaystyle{ \int _{0}^{{\pi}} x\cos nxdx } }$

$3$\displaystyle{ =\frac{\hspace{2}2\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\displaystyle{ \int _{0}^{{\pi}} x{ $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\sin nx $ }^{\prime }dx } }$

$3$\displaystyle{ =\frac{\hspace{2}2\hspace{2}}{\hspace{2}{\pi}\hspace{2}} \{ \[ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}x\sin nx \] _{0}^{{\pi}}- \displaystyle{ \int _{0}^{{\pi}} 1\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\sin nxdx } \} }$

$3$\displaystyle{ =\frac{\hspace{2}2\hspace{2}}{\hspace{2}{\pi}\hspace{2}} \{ 0- \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}} \[ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\cos nx \] _{0}^{{\pi}} \} }$

$3$\displaystyle{ =\frac{\hspace{2}2\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2} {n}^{2} \hspace{2}} $ \cos n{\pi}- 1 $ }$

$3$\displaystyle{ =\frac{\hspace{2}2\hspace{2}}{\hspace{2} {\pi}{n}^{2} \hspace{2}} \{ { $ - 1 $ }^{n}- 1 \} }$

$3$\displaystyle{ = \{ \begin{array}0& \vspace{6}\\ - \frac{\hspace{2}4\hspace{2}}{\hspace{2} {\pi}{n}^{2} \hspace{2}} & \vspace{6}\\ \end{array} \begin{array} & \vspace{6}\\ & \vspace{6}\\ \end{array}\begin{array} & \vspace{6}\\ & \vspace{6}\\ \end{array}\begin{array} $ n\text{が偶数のとき} $ & \vspace{6}\\ $ n\text{が奇数のとき} $ & \vspace{6}\\ \end{array} }$
MathMLのtextタグの中に日本語が含まれており、日本語が文字化けしています。今後直していきます。

$3$\displaystyle{ \hspace{1}{b}_{n}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\displaystyle{ \int _{ - {\pi} }^{{\pi}} f $x$ \sin nxdx } }$

$3$\displaystyle{ f $x$ }$ が偶関数なので、 $3$\displaystyle{ f $x$ \sin nxdx }$ は奇関数である(奇関数の積分を参照) ．よって

$3$ =0$

以上より

$3$\displaystyle{ f $x$ \sim \hspace{1}{a}_{0}+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} $ \hspace{1}{a}_{n}\cos nx+\hspace{1}{b}_{n}\sin nx $ } }$

$3$\displaystyle{ \sim \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{\pi}+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \frac{\hspace{2}2\hspace{2}}{\hspace{2} {\pi}{n}^{2} \hspace{2}} \{ { $ - 1 $ }^{n}- 1 \} \cos nx } }$

$3$\displaystyle{ \sim \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{\pi}- \frac{\hspace{2}4\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\left({ \cos x+ \frac{\hspace{2}1\hspace{2}}{\hspace{2}{3}^{2}\hspace{2}}\cos 3x+ \frac{\hspace{2}1\hspace{2}}{\hspace{2}{5}^{2}\hspace{2}}\cos 5x+ \frac{\hspace{2}1\hspace{2}}{\hspace{2}{7}^{2}\hspace{2}}\cos 7x+ \cdots }\right) }$

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最終更新日： 2023年7月7日