undergoe8m

2022-11-21

Show ${\int }_{s}^{\mathrm{\infty }}f\left(x\right)dx=\mathcal{L}\left\{\frac{F\left(t\right)}{t}\right\}$ given $f\left(x\right)={\int }_{0}^{\mathrm{\infty }}{e}^{-xt}F\left(t\right)dt$

apopihvj

Expert

Observe that
$\frac{{e}^{-st}}{t}={\int }_{s}^{\mathrm{\infty }}{e}^{-xt}dx.$
Then
${\int }_{0}^{\mathrm{\infty }}{e}^{-st}\frac{F\left(t\right)}{t}dt={\int }_{0}^{\mathrm{\infty }}{\int }_{s}^{\mathrm{\infty }}{e}^{-xt}dxF\left(t\right)dt={\int }_{s}^{\mathrm{\infty }}{\int }_{0}^{\mathrm{\infty }}{e}^{-xt}F\left(t\right)dtdx={\int }_{s}^{\mathrm{\infty }}f\left(x\right)dx,$
where in the second equality I have assumed that F is nice enough so that you can exchange the order of integration.

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