周期２πのフーリエ級数の公式

周期 $3$\displaystyle{}$ 2 π の周期関数 $3$\displaystyle{f $x$ }$ は

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

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

$3$\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 }$ )　･･････(2)

$3$\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 }$ )　･･････(3)

のようり三角関数の級数として表される．この級数のことを関数 $3$\displaystyle{f $x$ }$ のフーリエ級数といい， $3$\displaystyle{ \hspace{1}{a}_{0} }$ ， $3$\displaystyle{ \hspace{1}{a}_{n} }$ ， $3$\displaystyle{ \hspace{1}{b}_{n} }$ のことをフーリエ係数という．

関数 $3$\displaystyle{f $x$ }$ が不連続となる $3$x$ などでは $3$\displaystyle{f $x$ }$ の値とフーリエ級数の値が一致しない場合があるので， $3$\displaystyle{f $x$ }$ のフーリエ級数を表す場合

$3$\displaystyle{f $x$ = \hspace{1}{a}_{0} +{\sum }\limits_{ n=1 }^{\infty} $ \hspace{1}{a}_{n}\cos nx+\hspace{1}{b}_{n}\sin nx $ }$ ･･････(4)

のように「＝」を使わず，「～」を使い

$3$\displaystyle{f $x$ \sim \hspace{1}{a}_{0} +{\sum }\limits_{ n=1 }^{\infty} $ \hspace{1}{a}_{n}\cos nx+\hspace{1}{b}_{n}\sin nx $ }$ ･･････(5)

と表すのが一般的である．

■フーリエ係数を求める計算

● $3$\displaystyle{ \hspace{1}{a}_{0} }$ を求める計算

(4)の両辺を区間 $3$\displaystyle{ \[ - {\pi},{\pi} \] }$ で積分する．

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

$3$\displaystyle{\displaystyle{ \int _{ - {\pi} }^{{\pi}} f $x$ dx }=\displaystyle{ \int _{ - {\pi} }^{{\pi}} \hspace{1}{a}_{0}dx }}$ $3$\displaystyle{+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \displaystyle{ \int _{ - {\pi} }^{{\pi}} \hspace{1}{a}_{n}\cos nxdx } }}$ $3$\displaystyle{+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \displaystyle{ \int _{ - {\pi} }^{{\pi}} \hspace{1}{b}_{n}\sin nxdx } }}$

$3$\displaystyle{\displaystyle{ \int _{ - {\pi} }^{{\pi}} f $x$ dx }=\hspace{1}{a}_{0}\displaystyle{ \int _{ - {\pi} }^{{\pi}} dx }}$ $3$\displaystyle{+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \hspace{1}{a}_{n}\displaystyle{ \int _{ - {\pi} }^{{\pi}} \cos nxdx } }}$ $3$\displaystyle{+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \hspace{1}{b}_{n}\displaystyle{ \int _{ - {\pi} }^{{\pi}} \sin nxdx } }}$

ここで

$3$\displaystyle{\displaystyle{ \int _{ - {\pi} }^{{\pi}} dx }= \[x\] _{ - {\pi} }^{{\pi}}={\pi}+{\pi}=2{\pi}}$

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

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

以上より

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

これより，(1)の

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

が得られる．

● $3$\displaystyle{ \hspace{1}{a}_{n} }$ を求める計算

(4)の両辺に $3$\displaystyle{\cos mx}$ を掛けてから両辺を $3$\displaystyle{ \[ - {\pi},{\pi} \] }$ で積分する．

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

$3$\displaystyle{\displaystyle{ \int _{ - {\pi} }^{{\pi}} f $x$ \cos mxdx }}$

$3$\displaystyle{=\displaystyle{ \int _{ - {\pi} }^{{\pi}} \hspace{1}{a}_{0}\cos mxdx }}$ $3$\displaystyle{+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \displaystyle{ \int _{ - {\pi} }^{{\pi}} \hspace{1}{a}_{n}\cos nx\cos mxdx } }}$ $3$\displaystyle{+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \displaystyle{ \int _{ - {\pi} }^{{\pi}} \hspace{1}{b}_{n}\sin nx\cos mxdx } }}$

$3$\displaystyle{\displaystyle{ \int _{ - {\pi} }^{{\pi}} f $x$ \cos mxdx }}$

$3$\displaystyle{=\hspace{1}{a}_{0}\displaystyle{ \int _{ - {\pi} }^{{\pi}} \cos mxdx }}$ $3$\displaystyle{+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \hspace{1}{a}_{n}\displaystyle{ \int _{ - {\pi} }^{{\pi}} \cos nx\cos mxdx } }}$ $3$\displaystyle{+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \hspace{1}{b}_{n}\displaystyle{ \int _{ - {\pi} }^{{\pi}} \sin nx\cos mxdx } }}$

ここで

$3$\displaystyle{\displaystyle{ \int _{ - {\pi} }^{{\pi}} \cos mxdx }= \[ \frac{\hspace{2}1\hspace{2}}{\hspace{2}m\hspace{2}}\sin mx \] _{ - {\pi} }^{{\pi}}}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}m\hspace{2}}\sin m{\pi}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}m\hspace{2}}\sin m $ - {\pi} $ =0}$

$3$\displaystyle{\displaystyle{ \int _{ - {\pi} }^{{\pi}} \cos nx\cos mxdx }}$ の計算は， $3$\displaystyle{n\neq m}$ のとき，積和の公式を利用して

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

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

$3$\displaystyle{=0}$

$3$\displaystyle{n=m}$ のとき，半角の公式を利用して

$3$\displaystyle{=\displaystyle{ \int _{ - {\pi} }^{{\pi}} { \cos }^{2}mxdx }}$ $3$\displaystyle{=\displaystyle{ \int _{ - {\pi} }^{{\pi}} \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ 1+\cos 2mx $ dx }}$ $3$\displaystyle{= \[ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ x+\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2m \hspace{2}}\sin 2mx $ \] _{ - {\pi} }^{{\pi}}}$ $3$\displaystyle{={\pi}}$

$3$\displaystyle{\displaystyle{ \int _{ - {\pi} }^{{\pi}} \sin nx\cos mxdx }}$ の計算は， $3$\displaystyle{ \sin nx\cos mx }$ が奇関数であることより，0となる．

以上より

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

となり， $3$m$ を $3$n$ に書き換えて(2)の

$3$\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=0,1,2,\cdot \cdot \cdot \cdot \cdot \cdot \cdot $ }$

が得られる．

● $3$\displaystyle{ \hspace{1}{b}_{n} }$ を求める計算

(4)の両辺に $3$\displaystyle{\sin mx}$ を掛けてから両辺を $3$\displaystyle{ \[ - {\pi},{\pi} \] }$ で積分する．

$3$\displaystyle{\displaystyle{ \int _{ - {\pi} }^{{\pi}} f $x$ \sin mxdx }}$

$3$\displaystyle{=\displaystyle{ \int _{ - {\pi} }^{{\pi}} \hspace{1}{a}_{0}\sin mxdx }}$ $3$\displaystyle{+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \displaystyle{ \int _{ - {\pi} }^{{\pi}} \hspace{1}{a}_{n}\cos nx\sin mxdx } }}$ $3$\displaystyle{+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \displaystyle{ \int _{ - {\pi} }^{{\pi}} \hspace{1}{b}_{n}\sin nx\sin mxdx } }}$

$3$\displaystyle{\displaystyle{ \int _{ - {\pi} }^{{\pi}} f $x$ \sin mxdx }}$

$3$\displaystyle{=\hspace{1}{a}_{0}\displaystyle{ \int _{ - {\pi} }^{{\pi}} \sin mxdx }}$ $3$\displaystyle{+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \hspace{1}{a}_{n}\displaystyle{ \int _{ - {\pi} }^{{\pi}} \cos nx\sin mxdx } }}$ $3$\displaystyle{+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \hspace{1}{b}_{n}\displaystyle{ \int _{ - {\pi} }^{{\pi}} \sin nx\sin mxdx } }}$

ここで

$3$\displaystyle{\displaystyle{ \int _{ - {\pi} }^{{\pi}} \sin mxdx }= \[ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}m\hspace{2}}\cos mx \] _{ - {\pi} }^{{\pi}}}$

$3$\displaystyle{=- \frac{\hspace{2}1\hspace{2}}{\hspace{2}m\hspace{2}}\cos m{\pi}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}m\hspace{2}}\cos m $ - {\pi} $ }$

$3$\displaystyle{=1- 1=0}$

$3$\displaystyle{ \displaystyle{ \int _{ - {\pi} }^{{\pi}} \cos nx\sin mxdx } }$ の計算は， $3$\displaystyle{ \cos nx\sin mx }$ が奇関数より，0となる．

$3$\displaystyle{ \displaystyle{ \int _{ - {\pi} }^{{\pi}} \sin nx\sin mxdx } }$ の計算は， $3$\displaystyle{n\neq m}$ のとき

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

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

$3$\displaystyle{=0}$

$3$\displaystyle{n=m}$ のとき

$3$\displaystyle{=\displaystyle{ \int _{ - {\pi} }^{{\pi}} { \sin }^{2}mxdx }}$ $3$\displaystyle{=\displaystyle{ \int _{ - {\pi} }^{{\pi}} \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ 1- \cos 2mx $ dx }}$ $3$\displaystyle{= \[ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ x- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 2m \hspace{2}}\sin 2mx $ \] _{ - {\pi} }^{{\pi}}}$ $3$\displaystyle{={\pi}}$

以上より

$3$\displaystyle{\displaystyle{ \int _{ - {\pi} }^{{\pi}} f $x$ \sin mxdx }={\pi}\hspace{1}{b}_{m}}$

となり， $3$m$ を $3$n$ に書き換えて(3)の

$3$\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,\cdot \cdot \cdot \cdot \cdot \cdot \cdot $ }$

が得られる．

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最終更新日： 2026年1月12日

周期２πのフーリエ級数の公式

■フーリエ係数を求める計算

● を求める計算

● を求める計算

● を求める計算

● $3$\displaystyle{ \hspace{1}{a}_{0} }$ を求める計算

● $3$\displaystyle{ \hspace{1}{a}_{n} }$ を求める計算

● $3$\displaystyle{ \hspace{1}{b}_{n} }$ を求める計算