フーリエ級数の問題

■問題

$3$\displaystyle{ f\left({x}\right)=\left{{ \begin{array} - 1 & \left({ - {\pi}\leq x< 0 }\right) & \vspace{6}\\ 1& \left({ 0\leq x< {\pi} }\right) & \vspace{6}\\ \end{array}\text{\hspace{5}} }$ ， $3$\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\left({x}\right)\sim \frac{\hspace{2}4\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\left({ \sin x+\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\sin 3x+\frac{\hspace{2}1\hspace{2}}{\hspace{2}5\hspace{2}}\sin 5x+\frac{\hspace{2}1\hspace{2}}{\hspace{2}7\hspace{2}}\sin 7x+\cdot \cdot \cdot }\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{ \hspace{1}{a}_{0}=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2{\pi} \hspace{2}}\displaystyle{ \int _{ - {\pi} }^{{\pi}} f\left({x}\right)dx } }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2{\pi} \hspace{2}}\left({ \displaystyle{ \int _{ - {\pi} }^{0} \left({ - 1 }\right)dx }+\displaystyle{ \int _{0}^{{\pi}} dx } }\right) }$

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

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2{\pi} \hspace{2}}\left({ - {\pi}+{\pi} }\right) }$

$3$\displaystyle{ =0 }$

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

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\left({ \displaystyle{ \int _{ - {\pi} }^{0} \left({ - 1 }\right)\cos nxdx }+\displaystyle{ \int _{0}^{{\pi}} \cos nxdx } }\right) }$

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

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\left({ 0+0 }\right) }$

$3$\displaystyle{ =0 }$

$3$\displaystyle{ \hspace{1}{b}_{n}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\displaystyle{ \int _{ - {\pi} }^{{\pi}} f\left({x}\right)\sin nx }dx }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\left({ \displaystyle{ \int _{ - {\pi} }^{0} \left({ - 1 }\right)\sin nxdx }+\displaystyle{ \int _{0}^{{\pi}} \sin nxdx } }\right) }$

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

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\left[{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\left{{ 1- \cos \left({ - n{\pi} }\right) }\right}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\left{{ 1- \cos n{\pi} }\right} }\right] }$

余弦関数は偶関数なので $3$\displaystyle{ \cos \left({ - n{\pi} }\right)=\cos n{\pi} }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\left[{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\left{{ 1- \cos n{\pi} }\right}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\left{{ 1- \cos n{\pi} }\right} }\right] }$

$3$\displaystyle{ =\frac{\hspace{2} 2\left({ 1- \cos n{\pi} }\right) \hspace{2}}{\hspace{2} n{\pi} \hspace{2}} }$

$3$\displaystyle{ =\frac{\hspace{2} 2\left{{ 1- { \left({ - 1 }\right) }^{n} }\right} \hspace{2}}{\hspace{2} n{\pi} \hspace{2}} }$ 　（∵ $3$\displaystyle{ \cos n{\pi}={ \left({ - 1 }\right) }^{n} }$ ）

$3$\displaystyle{ =\left{{ \begin{array}0& \left({ n\text{が偶数} }\right) & \vspace{6}\\ \displaystyle{\frac{\hspace{2}4\hspace{2}}{\hspace{2} n{\pi} \hspace{2}}} & \left({ n\text{が奇数} }\right) & \vspace{6}\\ \end{array}\text{ } }$
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以上より

となる．

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最終更新日： 2023年7月7日