第9章 9.16

(1)

$\text{左辺} = \frac{1}{\alpha}\int_0^{\frac{\pi}{2}} \{\sin(\alpha x + wt) + \sin(\alpha x - wt)\}\,dx = \frac{1}{\alpha}\int_0^{\frac{\pi}{2}} 2\sin(\alpha x)\cdot\cos(wt)\,dx = \frac{2}{\alpha}\cos(wt)\int_0^{\frac{\pi}{2}}\sin(\alpha x)\,dx$

$= f_{(\alpha)}\cos(wt) = \text{右辺}$

これより

$f_{(\alpha)} = \frac{2}{\alpha}\int_0^{\frac{\pi}{2}}\sin(\alpha x)\,dx = \frac{2}{\alpha}\cdot\frac{1}{\alpha}\left[-\cos(\alpha x)\right]_0^{\frac{\pi}{2}} = \frac{2}{\alpha^2}\left(-\cos\left(\frac{\alpha}{2}\pi\right)+\cos 0\right) = \frac{2}{\alpha^2}\left(1-\cos\left(\frac{\alpha}{2}\pi\right)\right)$

(2)

$\lim_{\alpha\to 0} f_{(x)} = \lim_{\alpha\to 0}\frac{2}{\alpha^2}\left(1-\cos\left(\frac{\alpha}{2}\pi\right)\right) = \lim_{\alpha\to 0}\frac{2}{\alpha^2}\cdot 2\cdot\sin^2\frac{\alpha}{4}\pi = \lim_{\alpha\to 0}\left(\frac{2}{\alpha}\sin\left(\frac{\alpha}{4}\pi\right)\right)^2$

$\frac{\alpha}{4}\pi = \beta$

とおくと

$= \lim_{\beta\to 0}\left(\frac{\pi}{2\beta}\sin\beta\right)^2 = \lim_{\beta\to 0}\frac{\pi^2}{4}\left(\frac{\sin\beta}{\beta}\right)^2 = \frac{\pi^2}{4}$