フーリエ変換の問題

■問題

次の関数 ${\displaystyle f\left(x\right)}$ が，偶関数か奇関数かを判別し，偶関数ならば関数 ${\displaystyle f\left(x\right)}$ のフーリエ余弦変換 ${\displaystyle F_c\left(\omega \right)}$ を，奇関数ならば関数 ${\displaystyle f\left(x\right)}$ のフーリエ正弦変換 ${\displaystyle F_s\left(\omega \right)}$ を求めよ．

${\displaystyle f\left(x\right)=\{\begin{array}{cc}\left|x\right|& \left(-1<x<1\right)\\ 0& \left(\left|x\right|\geqq 1\right)\end{array}}$

■解答

${\displaystyle F_c\left(\omega \right)=\{\begin{array}{cc}\sqrt{\frac{2}{\pi }}\left\{\frac{1}{\omega }\sin \omega +\frac{1}{{\omega }^2}\left(\cos \omega -1\right)\right\}& \left(\omega \ne 0\right)\\ \frac{1}{\sqrt{2\pi }}& \left(\omega =0\right)\end{array}}$

■解き方

${\displaystyle f\left(x\right)}$ は偶関数である．よって，フーリエ余弦変換 ${\displaystyle F_c\left(\omega \right)}$ を求める．

${\displaystyle F_c\left(\omega \right)=\sqrt{\frac{2}{\pi }}{\displaystyle {\int }_0^{\infty }f\left(x\right)\cos \omega xdx}}$

${\displaystyle =\sqrt{\frac{2}{\pi }}{\displaystyle {\int }_0^{1}x\cos \omega xdx}}$

◇ $\omega \ne 0$ のとき

部分積分法を用いる

${\displaystyle =\sqrt{\frac{2}{\pi }}{\displaystyle {\int }_0^{1}x{\left(\frac{1}{\omega }\sin \omega x\right)}^{\prime }dx}}$

${\displaystyle =\sqrt{\frac{2}{\pi }}\left\{{\left[\frac{1}{\omega }x\sin \omega x\right]}_0^1-{\displaystyle {\int }_0^1\frac{1}{\omega }\sin \omega xdx}\right\}}$

${\displaystyle =\sqrt{\frac{2}{\pi }}\left\{\frac{1}{\omega }\sin \omega -\frac{1}{\omega }{\left[-\frac{1}{\omega }\cos \omega x\right]}_0^1\right\}}$

${\displaystyle =\sqrt{\frac{2}{\pi }}\left\{\frac{1}{\omega }\sin \omega -\frac{1}{\omega }\left(-\frac{1}{\omega }\cos \omega +\frac{1}{\omega }\right)\right\}}$

${\displaystyle =\sqrt{\frac{2}{\pi }}\left\{\frac{1}{\omega }\sin \omega +\frac{1}{{\omega }^2}\left(\cos \omega -1\right)\right\}}$

◇ $\omega ＝0$ のとき

${\displaystyle =\sqrt{\frac{2}{\pi }}{{\displaystyle \int }}_0^{1}xdx}$

${\displaystyle =\sqrt{\frac{2}{\pi }}{\left[\frac{1}{2}{x}^2\right]}_0^1}$

${\displaystyle =\sqrt{\frac{2}{\pi }}\cdot \frac{1}{2}}$

${\displaystyle =\frac{1}{\sqrt{2\pi }}}$

●備考

${\displaystyle \underset{\omega \to 0}{\lim }{F}_s\left(\omega \right)}$ ${\displaystyle =\underset{\omega \to 0}{\lim }\sqrt{\frac{2}{\pi }}\left\{\frac{1}{\omega }\sin \omega +\frac{1}{{\omega }^2}\left(\cos \omega -1\right)\right\}}$

${\displaystyle =\sqrt{\frac{2}{\pi }}\underset{\omega \to 0}{\lim }\left\{\frac{1}{\omega }\sin \omega +\frac{1}{{\omega }^2}\left(\cos \omega -1\right)\right\}}$

${\displaystyle =\sqrt{\frac{2}{\pi }}\left\{\underset{\omega \to 0}{\lim }\frac{\sin \omega }{\omega }+\underset{\omega \to 0}{\lim }\frac{\cos \omega -1}{{\omega }^2}\right\}}$

${\displaystyle =\sqrt{\frac{2}{\pi }}\left(1-\frac{1}{2}\right)}$

(∵ ${\displaystyle \underset{\omega \to 0}{\lim }\frac{\cos \omega -1}{{\omega }^2}=\underset{\omega \to 0}{\lim }\frac{-\sin \omega }{2\omega }}$ ${\displaystyle =-\frac{1}{2}}$   (ロピタルの定理を使っている) )

${\displaystyle =\frac{1}{\sqrt{2\pi }}}$

　

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最終更新日： 2026年9月29日