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August 20th, 2015, 04:54 PM   #1
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Inverse Laplace Transform/Complex Integral

I found what appears to be a definition of the inverse Laplace Transform which is as follows:

$\displaystyle f(t)=\frac{1} {2\pi i} \lim\limits_{T\to\infty} \int_{\sigma-iT}^{\sigma+iT} e^{st} \mathscr L \{f(t)\} \, ds$

I began trying to prove that the above is true.

This reduces to proving that $\displaystyle f(t)=\frac{1} {2\pi i} \lim\limits_{T\to\infty} \int_{\sigma-iT}^{\sigma+iT} \int_0^\infty f(t)\,dt\,ds$

I don't know where to continue from here. I know I'm done by the Cauchy Integral Formula if I can prove that $\displaystyle\lim\limits_{T\to\infty} \int_{\sigma-iT}^{\sigma+iT}\int_0^\infty f(t)\,dt\,ds=\oint_{\gamma} \frac{f'(z)} {z-t}\,dz$ where $\gamma$ is some contour in the complex plane whose interior region contains $t$ and is traversed counter-clockwise.

However, at the moment I am stuck and do not know how to continue. Any thoughts?

Note: The definition I am using for the Laplace Transform is $\displaystyle \mathscr L\{f(t)\}=\int_{0}^\infty e^{-st} f(t)\,dt$
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