複素フーリエ変換

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

${\displaystyle F\left(\omega \right)=\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_0^{\infty }f\left(x\right)e^{i\omega x}dx}}$

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

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

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

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

をオイラーの公式

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

を使って，複素数にまで拡張する．

${\displaystyle \cos wv=\frac{e^{iwv}+e^{-iwv}}{2}}$　･･････(2)

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

(1-1)に(2)を代入する．

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

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

(1-2)に(3)を代入する．

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

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

${\displaystyle \cos wx=\frac{e^{iwx}+e^{-iwx}}{2}}$　･･････(6)

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

(1)に(4)，(5)，(6)，(7)を代入する．

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

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

${\displaystyle -i\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{iwv}dv-\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{-iwv}du}}\right)\left(-i\frac{e^{iwx}-e^{-iwx}}{2}\right)\}dw}$

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

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

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

${\displaystyle +\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{iwv}dv}\right)e^{-iwx}-\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{-iwv}dv}\right)e^{-iwx}\}dw}$

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

${\displaystyle ={\displaystyle {\int }_0^{\infty }\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{-iwv}dv}\right)e^{iwx}dw}+{\displaystyle {\int }_0^{\infty }\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{iwv}dv}\right)e^{-iwx}dw}}$　･･････(8)

となる．(8)の右辺の第１項

${\displaystyle {\displaystyle {\int }_0^{\infty }\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{-iwv}dv}\right)e^{iwx}dw}}$　･･････(9)

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

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

${\displaystyle w}$ 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 ${\displaystyle w}$ に書き換える(${\displaystyle \gamma =w}$ の置換積分をする)と

${\displaystyle ={\displaystyle {\int }_{-\infty }^0\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{iwv}dv}\right)e^{-iwx}dw}}$　･･････(10)

(10)を(8)に代入すると

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

${\displaystyle {\displaystyle ={\int }_{-\infty }^{\infty }\left(\frac{1}{2\pi }f\left(v\right)e^{iwv}dv\right)e^{-iwx}dw}}$　･･････(11)

となる．

${\displaystyle \frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{-\infty }^{\infty }f\left(v\right)e^{iwv}dv=F\left(w\right)}}$　･･････(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(w\right)e^{-iwx}dw}$

${\displaystyle =\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{-\infty }^{\infty }F\left(w\right)e^{-iwx}dw}}$　･･････(13)

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

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

となる．

すなわち

${\displaystyle f\left(x\right)=\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{-\infty }^{\infty }F\left(w\right)e^{-iwx}dw}}$　･･････(13)

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

となる．

 

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最終更新日： 2023年7月3日