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 Calculus Calculus Math Forum

 September 15th, 2014, 03:05 AM #1 Newbie   Joined: Sep 2014 From: Adelaide Posts: 1 Thanks: 0 Laplace Transform I am usually not bad with laplace transform questions, but have not had any luck with this question: Find the laplace transform of (1-exp(-t))^(-1/2) Just any hints on where to start would be great! Thanks September 17th, 2014, 02:46 PM #2 Global Moderator   Joined: Dec 2006 Posts: 21,029 Thanks: 2259 According to W|A, it's $\displaystyle \frac{\sqrt\pi\Gamma(s)}{\Gamma(s+\frac12)}$. I'm not sure how to prove that by elementary means. September 29th, 2014, 09:22 AM #3 Senior Member   Joined: Dec 2013 From: some subspace Posts: 212 Thanks: 72 Math Focus: real analysis, vector analysis, numerical analysis, discrete mathematics $\displaystyle \mathcal{L} \left\{ \frac{1}{\sqrt{1-e^{-t}}} \right\} = \int_0 ^{\infty} \frac{e^{-st} \mathrm{d}t}{\sqrt{1-e^{-t}}}.$ Let's substitute $\displaystyle 1-e^{-t} = x^2 \quad \Rightarrow \quad \mathrm{d}t = \frac{2x\mathrm{d}x}{1-x^2}$. One gets $\displaystyle \mathcal{L} \left\{ \frac{1}{\sqrt{1-e^{-t}}} \right\} = 2\int_0 ^{1} \left( 1 - x^2 \right) ^{s - 1} \mathrm{d}x$. Then, after substituting $\displaystyle x = \sin \varphi$, the integral can be expressed in terms of beta function: $\displaystyle \mathcal{L} \left\{ \frac{1}{\sqrt{1-e^{-t}}} \right\} = 2\int_0 ^{\frac{\pi}{2}} \cos ^{2s-1} \varphi \, \mathrm{d}\varphi = 2\int_0 ^{\frac{\pi}{2}} \cos ^{2s-1} \varphi \, \sin ^{2\cdot \frac{1}{2} -1} \varphi \, \mathrm{d}\varphi = \beta \left( s, \frac{1}{2} \right) = \frac{\sqrt{\pi} \, \Gamma \left( s \right)}{\Gamma \left( s + \frac{1}{2} \right)}$. Thanks from ZardoZ and jks September 29th, 2014, 03:42 PM #4 Math Team   Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus @fysmat nice! Thanks from fysmat Tags laplace, transform 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 Deiota Calculus 1 April 28th, 2013 10:28 AM r-soy Calculus 2 March 24th, 2013 04:30 PM fredericP Calculus 6 January 1st, 2013 08:30 PM gelatine1 Calculus 5 November 2nd, 2012 12:17 PM perelachess Applied Math 0 March 24th, 2009 06:10 AM

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