フーリエ級数の問題

■問題

$3$\displaystyle{f\left({x}\right)=\left{{ \begin{array}2& \left({ - 2≦x< 0 }\right) & \vspace{6}\\ 2- x & \left({ 0≦x< 2 }\right) & \vspace{6}\\ \end{array} }$
TeXに変換設定していない数学記号や，特殊文字が含まれています。今後直していきます。
， $3$\displaystyle{ f $ x+4 $ =f $x$ }$

のフーリエ級数を求めよ．

■解答

$3$\displaystyle{ f $x$ \sim \frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}}+\frac{\hspace{2}4\hspace{2}}{\hspace{2} {{\pi}}^{2} \hspace{2}} $ \cos \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}x+\frac{\hspace{2}1\hspace{2}}{\hspace{2} {3}^{2} \hspace{2}}\cos \frac{\hspace{2} 3{\pi} \hspace{2}}{\hspace{2}2\hspace{2}}x+\frac{\hspace{2}1\hspace{2}}{\hspace{2} {5}^{2} \hspace{2}}\cos \frac{\hspace{2} 5{\pi} \hspace{2}}{\hspace{2}2\hspace{2}}x+\cdots \cdots $ }$ $3$\displaystyle{ +\frac{\hspace{2}2\hspace{2}}{\hspace{2}{\pi}\hspace{2}} $ - \sin \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}x+\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\sin \frac{\hspace{2} 2{\pi} \hspace{2}}{\hspace{2}2\hspace{2}}x- \frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\sin \frac{\hspace{2} 3{\pi} \hspace{2}}{\hspace{2}2\hspace{2}}x+\cdots \cdots $ }$

■ヒント

フーリエ係数

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

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

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

を求める

■解き方

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

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

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}} \{ 2 \[x\] _{ - 2 }^{0}+ \[ 2x- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{x}^{2} \] _{0}^{2} \} }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}} \{ 4+4- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\cdot 4 \} }$

$3$\displaystyle{ =\frac{\hspace{2}6\hspace{2}}{\hspace{2}4\hspace{2}} }$

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

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

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

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

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \{ \frac{\hspace{2}4\hspace{2}}{\hspace{2} n{\pi} \hspace{2}} $ 0- 0 $ + $ 0- 0 $ +\frac{\hspace{2}2\hspace{2}}{\hspace{2} n{\pi} \hspace{2}} \[ - \frac{\hspace{2}2\hspace{2}}{\hspace{2} n{\pi} \hspace{2}}\cos \frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}2\hspace{2}}x \] _{0}^{2} \} }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ - \frac{\hspace{2}4\hspace{2}}{\hspace{2} {n}^{2}{{\pi}}^{2} \hspace{2}} $ $ \cos n{\pi}- 1 $ }$

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

$3$\displaystyle{ =\left{{ \begin{array}0& \left({ n\text{が偶数のとき} }\right) & \vspace{6}\\ \displaystyle{ \frac{\hspace{2}4\hspace{2}}{\hspace{2} {n}^{2}{{\pi}}^{2} \hspace{2}} }& \left({ n\text{が奇数のとき} }\right) & \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}2\hspace{2}}\displaystyle{ \int _{ - l }^{l} f $x$ \sin \frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}2\hspace{2}} }xdx }$

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

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

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

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \{ - \frac{\hspace{2}4\hspace{2}}{\hspace{2} n{\pi} \hspace{2}} $ 1- { \( - 1 $ }^{n} \) +\frac{\hspace{2}4\hspace{2}}{\hspace{2} n{\pi} \hspace{2}}- 0 \} }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\cdot \frac{\hspace{2}4\hspace{2}}{\hspace{2} n{\pi} \hspace{2}}\cdot { $ - 1 $ }^{n} }$

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

$3$\displaystyle{ f $x$ \sim \hspace{1}{a}_{0}+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} $ \hspace{1}{a}_{n}\cos \frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}2\hspace{2}}+\hspace{1}{b}_{n}\sin \frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}2\hspace{2}}x $ } }$

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

$3$\displaystyle{ \sim \frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}}+\frac{\hspace{2}4\hspace{2}}{\hspace{2} {{\pi}}^{2} \hspace{2}} $ \cos \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}x+\frac{\hspace{2}1\hspace{2}}{\hspace{2} {3}^{2} \hspace{2}}\cos \frac{\hspace{2} 3{\pi} \hspace{2}}{\hspace{2}2\hspace{2}}x+\frac{\hspace{2}1\hspace{2}}{\hspace{2} {5}^{2} \hspace{2}}\cos \frac{\hspace{2} 5{\pi} \hspace{2}}{\hspace{2}2\hspace{2}}x+\cdots \cdots $ }$ $3$\displaystyle{ +\frac{\hspace{2}2\hspace{2}}{\hspace{2}{\pi}\hspace{2}} $ - \sin \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}2\hspace{2}}x+\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\sin \frac{\hspace{2} 2{\pi} \hspace{2}}{\hspace{2}2\hspace{2}}x- \frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\sin \frac{\hspace{2} 3{\pi} \hspace{2}}{\hspace{2}2\hspace{2}}x+\cdots \cdots $ }$

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最終更新日： 2023年7月6日