フーリエ変換の問題

■問題

偶関数

${\displaystyle f\left(x\right)=\{\begin{array}{cc}\frac{k}{L}& \left(-\frac{L}{2}<x<\frac{L}{2}\right)\\ 0& \left(\left|x\right|\geqq \frac{L}{2}\right)\end{array}}$

のフーリエ余弦変換 ${\displaystyle F_c\left(\omega \right)}$ を求めよ．また， ${\displaystyle \underset{L\to 0}{\lim }{F}_c\left(\omega \right)}$ を求めよ．

■解答

${\displaystyle F_c\left(\omega \right)=\{\begin{array}{cc}\frac{k}{L\omega }\sqrt{\frac{2}{\pi }}\sin \frac{L\omega }{2}& \left(\omega \ne 0\right)\\ \frac{k}{\sqrt{2\pi }}& \left(\omega =0\right)\end{array}}$

${\displaystyle \underset{L\to 0}{\lim }{F}_c\left(\omega \right)=\frac{k}{\sqrt{2\pi }}}$

■解き方

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

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

◇ $\omega \ne 0$ のとき

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

${\displaystyle =\sqrt{\frac{2}{\pi }}\cdot \frac{k}{L}\cdot \frac{1}{\omega }\sin \frac{L\omega }{2}}$

${\displaystyle =\frac{k}{L\omega }\sqrt{\frac{2}{\pi }}\sin \frac{L\omega }{2}}$

${\displaystyle \underset{L\to 0}{\lim }{F}_c\left(\omega \right)=\underset{L\to 0}{\lim }\frac{k}{L\omega }\sqrt{\frac{2}{\pi }}\sin \frac{L\omega }{2}}$

${\displaystyle =\underset{L\to 0}{\lim }\frac{k}{\sqrt{2\pi }}\cdot \frac{\sin \frac{L\omega }{2}}{\frac{L\omega }{2}}}$

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

（∵ここを参照）

◇ $\omega =0$ のとき

${\displaystyle =\sqrt{\frac{2}{\pi }}{{\displaystyle \int }}_0^{\frac{L}{2}}\frac{k}{L}dx}$

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

${\displaystyle =\sqrt{\frac{2}{\pi }}\cdot \frac{k}{L}\cdot \frac{L}{2}}$

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

${\displaystyle \underset{L\to 0}{\lim }{F}_c\left(\omega \right)=\underset{L\to 0}{\lim }\frac{k}{\sqrt{2\pi }}}$${\displaystyle =\frac{k}{\sqrt{2\pi }}}$

●備考

${\displaystyle \underset{\omega \to 0}{\lim }{F}_c\left(\omega \right)}$${\displaystyle =\underset{\omega \to 0}{\lim }\frac{k}{L\omega }\sqrt{\frac{2}{\pi }}\sin \frac{L\omega }{2}}$

${\displaystyle =\underset{\omega \to 0}{\lim }\frac{k}{L\omega }\sqrt{\frac{2}{\pi }}\cdot \frac{\sin \frac{L\omega }{2}}{\frac{L\omega }{2}}\cdot \frac{L\omega }{2}}$

${\displaystyle =\frac{k}{\sqrt{2\pi }}\left(\underset{\omega \to 0}{\lim }\frac{\sin \frac{L\omega }{2}}{\frac{L\omega }{2}}\right)}$

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

　

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最終更新日： 2026年8月13日