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\displaystyle \begin{aligned} w(t)=\int_0^{\infty}\frac{\sin tx}{x}\,dx\Rightarrow\mathcal{L}\{w(t)\} &=\int_0^{\infty}e^{-st}\int_0^{\infty}\frac{\sin tx}{x}\,dx\,dt\\&=\int_0^{\infty}\frac{1}{x}\int_0^{\infty}e^{-st}\sin tx\,dt\,dx\\&=\int_0^{\infty} \frac{1}{x}\left( \frac{x}{x^2+s^2} \right)\,dx \\&=\frac{\pi}{2s}
\tau_p = \displaystyle\sum_{k=0}^{p-1}\Big{(}\frac{k}{p}\Big{)} e^{\frac{2\pi i k}{p}}
\Big{(}\frac{k}{p}\Big{)}
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