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6.52 Proof of theorem

To prove the theorem, we observe:

\frac{d}{ds}\left(\int_{0}^{R}e^{-sx}f(x)\,dx\right)=\int_{0}^{R}\frac{% \partial}{\partial s}\left(e^{-sx}f(x)\right)\,dx.

Now

\frac{\partial}{\partial s}\left(e^{-sx}f(x)\right)=-xe^{-sx}f(x).

So the right-hand side converges to $-{\mathcal{L}}(xf(x))$ as $R\rightarrow\infty$ . Since the left-hand side converges to $\frac{dF}{ds}$ , this completes the proof.