
Complex Analysis Complex Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
August 20th, 2015, 04:54 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: $\displaystyle f(t)=\frac{1} {2\pi i} \lim\limits_{T\to\infty} \int_{\sigmaiT}^{\sigma+iT} e^{st} \mathscr L \{f(t)\} \, ds$ I began trying to prove that the above is true. This reduces to proving that $\displaystyle f(t)=\frac{1} {2\pi i} \lim\limits_{T\to\infty} \int_{\sigmaiT}^{\sigma+iT} \int_0^\infty f(t)\,dt\,ds$ I don't know where to continue from here. I know I'm done by the Cauchy Integral Formula if I can prove that $\displaystyle\lim\limits_{T\to\infty} \int_{\sigmaiT}^{\sigma+iT}\int_0^\infty f(t)\,dt\,ds=\oint_{\gamma} \frac{f'(z)} {zt}\,dz$ where $\gamma$ is some contour in the complex plane whose interior region contains $t$ and is traversed counterclockwise. 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  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Relatioship between Laplace and Inverse Laplace Transform from tables  szz  Differential Equations  1  November 2nd, 2014 04:18 AM 
ODE and Inverse Laplace Transform  szz  Calculus  1  November 1st, 2014 04:14 PM 
Inverse Laplace Transform  Bkalma  Calculus  1  May 6th, 2014 08:08 PM 
Laplace tranform and inverse of Laplace transform  Deiota  Calculus  1  April 28th, 2013 11:28 AM 
Inverse Laplace Transform  timh  Applied Math  1  June 7th, 2010 07:44 PM 