第9章 9.14

(1)

$\int_{-\pi}^{\pi} dx = [x]_{-\pi}^{\pi} = \pi - (-\pi) = 2\pi$

(2)

$\int_{-\pi}^{\pi} \sin(nx)\,dx = \frac{1}{n}[-\cos(nx)]_{-\pi}^{\pi} = \frac{1}{n}\{-\cos(n\pi) + \cos(-n\pi)\} = \frac{1}{n}\{-(-1)^n + (-1)^n\} = 0$

(3)

$\int_{-\pi}^{\pi} \cos(nx)\,dx = \frac{1}{n}[\sin(nx)]_{-\pi}^{\pi} = \frac{1}{n}\{\sin(n\pi) - \sin(-n\pi)\} = \frac{1}{n}(0-0) = 0$

(4)

m = n のとき

$\int_{-\pi}^{\pi} \sin(mx)\cdot\sin(mx)\,dx = \int_{-\pi}^{\pi} \frac{1-\cos 2mx}{2}\,dx = \frac{1}{2}\Big[x - \frac{1}{2m}\sin(2mx)\Big]_{-\pi}^{\pi} = \frac{1}{2}\{\pi-(-\pi)\} = \pi$

m ≠ n のとき

$\int_{-\pi}^{\pi} \sin(mx)\cdot\sin(nx)\,dx = \frac{1}{2}\int_{-\pi}^{\pi}\{\cos(m-n)-\cos(m+n)\}dx = \frac{1}{2}\left\{\int_{-\pi}^{\pi}\cos(m-n)\,dx - \int_{-\pi}^{\pi}\cos(m+n)\,dx\right\} = 0$

(5)

m = n のとき

$\int_{-\pi}^{\pi} \cos(mx)\cdot\cos(mx)\,dx = \int_{-\pi}^{\pi}\cos^2(mx)\,dx = \int_{-\pi}^{\pi}\frac{1+\cos(2mx)}{2}\,dx = \frac{1}{2}\left\{\int_{-\pi}^{\pi}dx - \int_{-\pi}^{\pi}\cos(2mx)\,dx\right\}$

$= \frac{1}{2}(2\pi - 0) = \pi$

m ≠ n のとき

$\int_{-\pi}^{\pi}\cos(mx)\cdot\cos(nx)\,dx = \frac{1}{2}\int_{-\pi}^{\pi}\{\cos(m-n)+\cos(m+n)\}dx = \frac{1}{2}(0-0) = 0$

(6)

$\int_{-\pi}^{\pi}\sin(mx)\cdot\cos(nx)\,dx = \frac{1}{2}\int_{-\pi}^{\pi}\{\sin(m-n)+\sin(m+n)\}dx = \frac{1}{2}(0-0) = 0$