tokoharuの落書き帳

らくがきですよ

京大の試験で出たらしい問題
 \lim_{x \to \infty} \int_{t=0}^{\infty} \frac{cos(xt)}{1+t^2}dt =0  をしめす
あってるか謎い概略

x=y \pi
(0\leq k < \frac{2}{y} )
 g(k) = \sum_{i=0}^{\infty } \frac{ cos((\frac{2i}{y})+k)y \pi )}{  1 + ( \frac{2i}{y}+k  )^{2} }
 =   cos(ky\pi) \sum_{i=0}^{\infty} \frac{ 1}{  1 + ( \frac{2i}{y}+k  )^{2} }
  A(k) = \sum_{i=0}^{\infty} \frac{ 1}{  1 + ( \frac{2i}{y}+k  )^{2}
  \sum_{i=1}^{\infty} \frac{ 1}{1+(\frac{2i}{y})^{2}}  \leq A(k) \leq \sum_{i=0}^{\infty} \frac{ 1}{1+( \frac{2i}{y})^{2}}
 -1 \leq A(k)-A(k+\frac{1}{y}) \leq 1

 -|cos(ky\pi)| \leq g(k) +g(k+\frac{1}{y}) \leq |cos(ky\pi)|