複素フーリエ級数

複素フーリエ級数は，フーリエ級数を複素数にまで拡張したもので

$3$\displaystyle{ f\left({x}\right)=\displaystyle{{\sum }\limits_{ n=- \infty }^{\infty} \hspace{1}{a}_{n}{e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x } } }$

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

と表される．

■導出

フーリエ級数

$3$\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}\ell \hspace{2}}x+\hspace{1}{b}_{n}\sin \frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x $ } }$ 　･･････(1)

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

$3$\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 }}$ 　･･････(1b)

$3$\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 }}$ 　･･････(1c)

$3$\displaystyle{ \left({ n=1,2,\cdots }\right) }$

をオイラーの公式

$3$\displaystyle{{e}^{ i{\theta} }=\cos {\theta}+i\sin {\theta}}$ 　･･････(2)

より，複素数にまで拡張する．(2)より

$3$\displaystyle{{e}^{ - i{\theta} }}$ $3$\displaystyle{=\cos $ - {\theta} $ +i\sin $ - {\theta} $ }$ $3$\displaystyle{=\cos {\theta}- i\sin {\theta}}$ 　･･････(3)

が得られる．(2)+(3)より

$3$\displaystyle{{e}^{ i{\theta} }+{e}^{ - i{\theta} }=2\cos {\theta}}$

$3$\displaystyle{\cos {\theta}=\frac{\hspace{2} {e}^{ i{\theta} }+{e}^{ - i{\theta} } \hspace{2}}{\hspace{2}2\hspace{2}}}$ 　･･････(4)

(2)-(3)より

$3$\displaystyle{ {e}^{ i{\theta} }- {e}^{ - i{\theta} }=2i\sin {\theta} }$

$3$\displaystyle{\sin {\theta}=\frac{\hspace{2} {e}^{ i{\theta} }- {e}^{ - i{\theta} } \hspace{2}}{\hspace{2} 2i \hspace{2}}}$ $3$\displaystyle{=\frac{\hspace{2} i $ {e}^{ i{\theta} }- {e}^{ - i{\theta} } $ \hspace{2}}{\hspace{2} 2{i}^{2} \hspace{2}}}$ $3$\displaystyle{=- i\frac{\hspace{2} {e}^{ i{\theta} }- {e}^{ - i{\theta} } \hspace{2}}{\hspace{2}2\hspace{2}}}$ 　･･････(5)

が得られる．

(4)より

$3$\displaystyle{\cos \frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x=\frac{\hspace{2} {e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }+{e}^{ - i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x } \hspace{2}}{\hspace{2}2\hspace{2}}}$ 　･･････(6)

(5)より

$3$\displaystyle{\sin \frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x=- i\frac{\hspace{2} {e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }- {e}^{ - i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x } \hspace{2}}{\hspace{2}2\hspace{2}}}$ 　･･････(7)

となる．

(1b)に(6)を代入する．

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

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2\ell \hspace{2}}\displaystyle{ \int _{ - \ell }^{\ell } f\left({x}\right){e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }dx }+\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2\ell \hspace{2}}\displaystyle{ \int _{ - \ell }^{\ell } f\left({x}\right){e}^{ - i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }dx }}$ 　･･････(8)

(8)の右辺の第2項を

$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2\ell \hspace{2}}{\displaystyle{ \int _{ - \ell }^{\ell } f $x$ e }}^{ - i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }dx=\hspace{1}{c}_{n}}$ 　･･････(9)

とおく．

(8)の辺の第1項は

$3$\displaystyle{ \frac{\hspace{2}1\hspace{2}}{\hspace{2} 2\ell \hspace{2}}\displaystyle{ \int _{ - \ell }^{\ell } f\left({x}\right){e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }dx }=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2\ell \hspace{2}}\displaystyle{ \int _{ - \ell }^{\ell } f\left({x}\right){e}^{ - i\frac{\hspace{2} \left({ - n }\right){\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }dx } }$ 　･･････(10)

となり，(9）の $3$\displaystyle{n}$ のところが $3$\displaystyle{- n}$ になっている．よって(8)式の右辺の第２項は $3$\displaystyle{\hspace{1}{c}_{ - n }}$ と表わすことができる．したがって，

$3$\displaystyle{\hspace{1}{a}_{n}=\hspace{1}{C}_{ - n }+\hspace{1}{C}_{n}}$ 　･･････(11)

となる．

同様にして(10)に(7)を代入する．

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

$3$\displaystyle{=- i $ \frac{\hspace{2}1\hspace{2}}{\hspace{2} 2\ell \hspace{2}}\displaystyle{ \int _{ - \ell }^{\ell } f \(x$ {e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }dx- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 2\ell \hspace{2}}\displaystyle{ \int _{ - \ell }^{\ell } f $x$ {e}^{ - i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }dx } } \) }$

$3$\displaystyle{=- i $ \hspace{1}{C}_{ - n }- \hspace{1}{C}_{n} $ }$ 　･･････(12)

となる．

$3$\displaystyle{\hspace{1}{a}_{0}=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2\ell \hspace{2}}\displaystyle{ \int _{ - \ell }^{\ell } f $x$ dx= }\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2\ell \hspace{2}}\displaystyle{ \int _{ - \ell }^{\ell } f $x$ {e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}\cdot 0 }= }\hspace{1}{C}_{0}}$ 　･･････(13)

となる．

(1)に(11)，(12)，(13)，(6)，(7)を代入すると

$3$\displaystyle{f $x$ \sim \hspace{1}{C}_{0}+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \{ $ \hspace{1}{C}_{ - n }+\hspace{1}{C}_{n} $ \frac{\hspace{2} {e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }+{e}^{ - i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x } \hspace{2}}{\hspace{2}2\hspace{2}} } - i $ \hspace{1}{C}_{ - n }- \hspace{1}{C}_{n} $ $ - i\frac{\hspace{2} {e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }- {e}^{ - i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x } \hspace{2}}{\hspace{2}2\hspace{2}} $ \} }$

$3$\displaystyle{ \sim \hspace{1}{C}_{0}+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\left{{ \hspace{1}{C}_{ - n }{e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }+\hspace{1}{C}_{n}{e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }+\hspace{1}{C}_{ - n }{e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }+\hspace{1}{C}_{n}{e}^{ - i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }- \hspace{1}{C}_{ - n }{e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }+\hspace{1}{C}_{n}{e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }+\hspace{1}{C}_{ - n }{e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }- \hspace{1}{C}_{n}{e}^{ - i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x } }\right} } }$

$3$\displaystyle{\sim \hspace{1}{C}_{0}+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} $ \hspace{1}{C}_{n}{e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }+\hspace{1}{C}_{ - n }{e}^{ - i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x } $ }}$

$3$\displaystyle{\sim \hspace{1}{C}_{0}+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \hspace{1}{C}_{n}{e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \hspace{1}{C}_{ - n }{e}^{ - i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }\) } }}$

$3$\displaystyle{\sim \hspace{1}{C}_{0}{e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}0 }+\displaystyle{{\sum }\limits_{ n=1 }^{\infty} \hspace{1}{C}_{n}{e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }+\displaystyle{{\sum }\limits_{ n=- \infty }^{ - 1 } \hspace{1}{C}_{n}{e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x } } }}$

$3$\displaystyle{\sim \displaystyle{{\sum }\limits_{ n=- \infty }^{\infty} \hspace{1}{C}_{n}{e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x } }}$

となる．すなわち

$3$\displaystyle{f $x$ \sim \displaystyle{{\sum }\limits_{ n=- \infty }^{\infty} \hspace{1}{C}_{n}{e}^{ i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x } }}$

$3$\displaystyle{\hspace{1}{C}_{n}=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2\ell \hspace{2}}\displaystyle{ \int _{ - \ell }^{\ell } f $x$ {e}^{ - i\frac{\hspace{2} n{\pi} \hspace{2}}{\hspace{2}\ell \hspace{2}}x }dx }}$

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最終更新日： 2023年7月3日