24/04/24 02:21:11.48 LloxEhQT.net
>>466
〔参考書〕
高木貞治「解析概論」改訂第三版、岩波書店 (1961)
第4章、§48．定理42．ｐ.166～167
>>467
F(1) = 0, (← 揚足取 御免)
>>468
(1) 和積公式より
sin(2kx) － sin(2(k-1)x) = 2sin(x)･cos((2k-1)x),
k = 1,2,…,n でたす。
(2) 積和公式より
4∫[0,π/2] cos((2i-1)x) cos(2j-1)x) dx
= 2∫[0,π/2] {cos(2(i+j-1)x) + cos(2(i-j)x)} dx
= 2∫[0,π/2] cos(2(i-j)x) dx
= δ_(i,j)・π,
i, j = 1,2,…,n でたす。
(3)
1/sin(x)^2－1 = 1/tan(x)^2 < 1/x^2 < 1/sin(x)^2,
を(2)に入れると
∫[0,π/2] (sin(2nx)/x)^2 dx = (n－θ/2)π (0<θ<1)
(4)
∫[0,∞] (sin(y)/y)^2 dy
= lim[n→∞] ∫[0,nπ] (sin(y)/y) dy
= lim[n→∞] (1/2n)∫[0,π/2] (sin(2nx)/x)^2 dx
= lim[n→∞] (π/2n) (n－θ/2) (0<θ<1)
= lim[n→∞] (π/2) (1－θ/2n)
= π/2.