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Question Number 190420 by mnjuly1970 last updated on 02/Apr/23

$$\mathrm{laplace}\mathrm{transform}$$$$-------$$$${\mathcal{L}}_t\left\{\frac{\sin \left(t\right)}{t}\right\}=\mathcal{F}\left(s\right)$$$$$$$$\mathcal{F}\left(s\right)=?$$$$$$$$thencalculate.$$$$$$$$\mathrm{\Omega }={\int }_0^{\mathrm{\infty }}\frac{e^{-2t}sin\left(t\right)}{t}dt=?$$$$--------$$

Commented by mahdipoor last updated on 02/Apr/23

$$$$$$L\left(\frac{sin\left(t\right)}{t}\right)=L\left(\frac{t-\frac{t^3}{3!}+\frac{t^5}{5!}-...}{t}\right)=L(1-\frac{t^2}{3!}+\frac{t^4}{5!}-...)$$$$=L\left(1\right)-\frac{1}{3!}L\left(t^2\right)+\frac{1}{5!}L\left(t^4\right)-...=$$$$\frac{1}{s}-\frac{2!}{3!s^3}+\frac{4!}{5!s^5}-....=\frac{1}{s}-\frac{1}{3s^3}+\frac{1}{5s^5}-...=G\left(s\right)$$$$G\left(s\right)=?\Rightarrow \mathrm{\Omega }=G\left(2\right)=?$$$$$$

Commented by mnjuly1970 last updated on 02/Apr/23

$$thx...G\left(s\right)=arctan\left(\frac{1}{s}\right)$$$$G\left(2\right)=arctan\left(\frac{1}{2}\right)...$$

Answered by witcher3 last updated on 04/Apr/23

$$\mathrm{F}\left(\mathrm{s}\right)=\mathrm{Im}{\int }_0^{\mathrm{\infty }}{\mathrm{t}}^{-1}{\mathrm{e}}^{-\mathrm{t}(\mathrm{s}-\mathrm{i})}\mathrm{dt},\mathrm{s}>0$$$${\mathrm{F}}^{\prime }\left(\mathrm{s}\right)=\mathrm{Im}{\int }_0^{\mathrm{\infty }}-{\mathrm{e}}^{-\mathrm{ts}+\mathrm{it}}\mathrm{dt}$$$$=\mathrm{Im}\frac{1}{-\mathrm{s}+\mathrm{i}}$$$$=-\frac{1}{{\mathrm{s}}^2+1}\Rightarrow \mathrm{F}\left(\mathrm{s}\right)=-\arctan \left(\mathrm{s}\right)+\mathrm{c}$$$$\underset{x\to \mathrm{\infty }}{\lim F}\left(\mathrm{x}\right)=0\Rightarrow \mathrm{c}=\frac{\pi }{2}$$$$\mathrm{\Omega }=-\arctan \left(2\right)+\frac{\pi }{2}=\arctan \left(\frac{1}{2}\right)$$$$$$$$$$$$$$

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