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 August 20th, 2015, 03:54 PM #1 Senior Member   Joined: Aug 2014 From: United States Posts: 137 Thanks: 21 Math Focus: Learning 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 $\displaystyle \mathscr L\{f(t)\}=\int_{0}^\infty e^{-st} f(t)\,dt$ Tags integral, inverse, laplace, transform or complex Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post szz Differential Equations 1 November 2nd, 2014 03:18 AM szz Calculus 1 November 1st, 2014 03:14 PM Bkalma Calculus 1 May 6th, 2014 07:08 PM Deiota Calculus 1 April 28th, 2013 10:28 AM timh Applied Math 1 June 7th, 2010 06:44 PM

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