2.3.7 $B:9J,2r$N87L)2r$X$N<}B+(B

$\displaystyle \left\vert u(x,t)-v(x,t)\right\vert$	$\displaystyle =$	$\displaystyle \left\vert \sum_{n=1}^\infty \sin n\pi x \left(f_n \cos n\pi t+g_... ...t(f_n \cos\beta_n t+g_n \frac{k\sin\beta_n t}{\sin\beta_n k}\right) \right\vert$
	$\displaystyle \le$	$\displaystyle \sum_{n=1}^\infty \left(\vert f_n\vert\left\vert\cos n\pi t-\cos\... ...ac{\sin n\pi t}{n\pi}-\frac{k\sin\beta_n t}{\sin \beta_n k} \right\vert \right)$
	$\displaystyle \le$	$\displaystyle \sum_{n=1}^N (\vert f_n\vert\left\vert\cos n\pi t-\cos\beta_n t\r... ...ert\frac{\sin n\pi t}{n\pi}-\frac{k\sin\beta_n t}{\sin \beta_n k} \right\vert )$
		$\displaystyle +\sum_{n>N} \vert f_n\vert(\vert\cos n\pi t\vert+\vert\cos\beta_n... ...right\vert +\left\vert\frac{k\sin \beta_n t}{\sin \beta_n k}\right\vert \right)$
	$\displaystyle \le$	$\displaystyle \sup_{1\le n\le N\atop \vert t\vert\le T}\vert\cos n\pi t-\cos\be... ...pi}-\frac{k\sin\beta_n t}{\sin\beta_n k} \right\vert\sum_{n=1}^N \vert g_n\vert$
		$\displaystyle +\sum_{n>N} \vert f_n\vert(\vert\cos n\pi t\vert+\vert\cos\beta_n... ...right\vert +\left\vert\frac{k\sin \beta_n t}{\sin \beta_n k}\right\vert \right)$
	$\displaystyle \le$	$\displaystyle M_f \sup_{1\le n\le N\atop \vert t\vert\le T}\vert\cos n\pi t-\co... ...t\vert\frac{\sin n\pi t}{n\pi}-\frac{k\sin\beta_n t}{\sin\beta_n k} \right\vert$
		$\displaystyle +\left(2+\frac{1}{\pi}+T\right)\eps$

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2.3.7 $B:9J,2r$N87L)2r$X$N<}B+(B