複素フーリエ変換

${\displaystyle f\left(x\right)=\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{-\infty }^{\infty }F\left(\omega \right)e^{-i\omega x}d\omega }}$

${\displaystyle F\left(\omega \right)=\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{-\infty }^{\infty }f\left(x\right)e^{i\omega x}dx}}$

となる． ${\displaystyle F\left(\omega \right)}$ を ${\displaystyle f\left(x\right)}$ のフーリエ変換という．

■導出

フーリエ積分

${\displaystyle f\left(x\right)={\displaystyle {\int }_0^{\infty }\left\{G\left(\omega \right)\cos \omega x+H\left(\omega \right)\sin \omega x\right\}d\omega }}$ ･･････(1)

${\displaystyle G\left(\omega \right)=\frac{1}{\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)\cos \omega vdv}}$ ･･････(1-1)

${\displaystyle H\left(\omega \right)=\frac{1}{\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)\sin \omega vdv}}$ ･･････(1-2)

をオイラーの公式

${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }$

を使って，複素指数関数を用いた形に書き換える．

${\displaystyle \cos \omega v=\frac{e^{i\omega v}+e^{-i\omega v}}{2}}$ ･･････(2)

${\displaystyle \sin \omega v=-i\frac{e^{i\omega v}-e^{-i\omega v}}{2}}$ ･･････(3)

(1-1)に(2)を代入する．

${\displaystyle G\left(\omega \right)}$ ${\displaystyle =\frac{1}{\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)\frac{e^{i\omega v}+e^{-i\omega v}}{2}dv}}$

${\displaystyle =\frac{1}{2\pi }{\int }_{-\infty }^{\infty }f\left(v\right)e^{i\omega v}dv+\frac{1}{2\pi }{\int }_{-\infty }^{\infty }f\left(v\right)e^{-i\omega v}dv}$ ･･････(4)

(1-2)に(3)を代入する．

${\displaystyle H\left(\omega \right)}$ ${\displaystyle =\frac{1}{\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)\left(-i\frac{e^{i\omega v}-e^{-i\omega v}}{2}\right)dv}}$

${\displaystyle =-i\left(\frac{1}{2\pi }{\int }_{-\infty }^{\infty }f\left(v\right)e^{i\omega v}dv-\frac{1}{2\pi }{\int }_{-\infty }^{\infty }f\left(v\right)e^{-i\omega v}dv\right)}$ ･･････(5)

${\displaystyle \cos \omega x=\frac{e^{i\omega x}+e^{-i\omega x}}{2}}$ ･･････(6)

${\displaystyle \sin \omega x=-i\frac{e^{i\omega x}-e^{-i\omega x}}{2}}$ ･･････(7)

(1)に(4)，(5)，(6)，(7)を代入する．

${\displaystyle f\left(x\right)=}$

${\displaystyle {\displaystyle {\int }_0^{\infty }\{\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{i\omega v}dv+\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{-i\omega v}dv}}\right)\frac{e^{i\omega x}+e^{-i\omega x}}{2}}}$

${\displaystyle -i\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{i\omega v}dv-\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{-i\omega v}dv}}\right)\left(-i\frac{e^{i\omega x}-e^{-i\omega x}}{2}\right)\}d\omega }$

${\displaystyle ={\displaystyle {\int }_0^{\infty }\frac{1}{2}\{\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{i\omega v}dv}\right)e^{i\omega x}+\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{-i\omega v}dv}\right)e^{i\omega x}}}$

${\displaystyle +\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{i\omega v}dv}\right)e^{-i\omega x}+\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{-i\omega v}dv}\right)e^{-i\omega x}}$

${\displaystyle -\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{i\omega v}dv}\right)e^{i\omega x}+\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{-i\omega v}dv}\right)e^{i\omega x}}$

${\displaystyle +\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{i\omega v}dv}\right)e^{-i\omega x}-\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{-i\omega v}dv}\right)e^{-i\omega x}\}d\omega }$

${\displaystyle ={\displaystyle {\int }_0^{\infty }\left\{\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{-i\omega v}dv}\right)e^{i\omega x}+\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{i\omega v}dv}\right)e^{-i\omega x}\right\}d\omega }}$

${\displaystyle ={\displaystyle {\int }_0^{\infty }\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{-i\omega v}dv}\right)e^{i\omega x}d\omega }+{\displaystyle {\int }_0^{\infty }\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{i\omega v}dv}\right)e^{-i\omega x}d\omega }}$ ･･････(8)

となる．(8)の右辺の第１項

${\displaystyle {\displaystyle {\int }_0^{\infty }\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{-i\omega v}dv}\right)e^{i\omega x}d\omega }}$ ･･････(9)

について， ${\displaystyle \omega =-\gamma }$ と置く置換積分をする．

${\displaystyle d\omega =-d\gamma }$

${\displaystyle \omega }$ が ${\displaystyle 0}$ から ${\displaystyle \infty }$ のとき， ${\displaystyle \gamma }$ は ${\displaystyle 0}$ から ${\displaystyle -\infty }$ となる．よって(9)は

${\displaystyle {\displaystyle {\int }_0^{-\infty }\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{i\gamma v}dv}\right)e^{-i\gamma x}\left(-d\gamma \right)}}$

${\displaystyle ={\displaystyle {\int }_{-\infty }^0\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{i\gamma v}dv}\right)e^{-i\gamma x}d\gamma }}$

となる．積分変数 ${\displaystyle \gamma }$ を ${\displaystyle \omega }$ に書き換える( ${\displaystyle \gamma =\omega }$ の置換積分をする)と

${\displaystyle ={\displaystyle {\int }_{-\infty }^0\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{i\omega v}dv}\right)e^{-i\omega x}d\omega }}$ ･･････(10)

(10)を(8)に代入すると

${\displaystyle f\left(x\right)=}$ ${\displaystyle {\displaystyle {\int }_{-\infty }^0\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{i\omega v}dv}\right)e^{-i\omega x}d\omega }}$ ${\displaystyle +{\displaystyle {\int }_0^{\infty }\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{i\omega v}dv}\right)}{e}^{-i\omega x}d\omega }$

${\displaystyle ={{\displaystyle \int }}_{-\infty }^{\infty }\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{i\omega v}dv}\right)e^{-i\omega x}d\omega }$ ･･････(11)

となる．

${\displaystyle \frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }f\left(v\right)e^{i\omega v}dv=F\left(\omega \right)}$ ･･････(12)

とおく．(12)を(11)に代入すると

${\displaystyle f\left(x\right)}$ ${\displaystyle ={\displaystyle {\int }_{-\infty }^{\infty }\frac{1}{\sqrt{2\pi }}}F\left(\omega \right)e^{-i\omega x}d\omega }$

${\displaystyle =\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{-\infty }^{\infty }F\left(\omega \right)e^{-i\omega x}d\omega }}$ ･･････(13)

となる．(12)の積分変数を ${\displaystyle v}$ から ${\displaystyle x}$ に書き換えると

${\displaystyle F\left(\omega \right)=\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{-\infty }^{\infty }f\left(x\right)e^{i\omega x}dx}}$ ･･････(14)

となる．

すなわち

${\displaystyle f\left(x\right)=\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{-\infty }^{\infty }F\left(\omega \right)e^{-i\omega x}d\omega }}$

${\displaystyle F\left(\omega \right)=\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{-\infty }^{\infty }f\left(x\right)e^{i\omega x}dx}}$

となる．

 

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最終更新日： 2026年9月29日