Proving the Inverse Laplace Transform of 1/(sqrt(s)+a)

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Discussion Overview

The discussion revolves around the inverse Laplace transform of the function 1/(sqrt(s)+a). Participants are exploring methods to prove the transform and discussing challenges related to integration and the presence of the square root in the denominator.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in proving the inverse Laplace transform and mentions an integral representation involving a contour integral.
  • Another participant notes the absence of a good antiderivative for the expression e^(st)/(sqrt(s)+a) and invites further insights on this matter.
  • A participant suggests that the result obtained from Mathematica is expressed in terms of error functions and seeks guidance on transforming it into that form.
  • Another participant proposes that e^(st) could be expressed as e^((sqrt(s)^2)*t), indicating that this might relate to the error function and serve as a potential approach to the problem.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the best method to approach the problem, and multiple competing views and methods are presented without resolution.

Contextual Notes

There are unresolved mathematical steps regarding the integration process and the transformation into error functions, as well as dependencies on the definitions of the functions involved.

iiternal
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Hi, all.
I am doing an inverse Laplace transform and meeting some difficulty.
InverseLaplaceTransform[1/(Sqrt+a)].
Using Mathematica I found the result, however, I failed to prove it. I know it should be like
\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty} \frac{1}{\sqrt{s}+a}ds
and I can take \gamma =0.
But, then, I cannot continue. The sqrt on the denominator killed me.
Thank you very much.
 
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From what I can tell, there is no good antiderivative of e^(st)/(Sqrt+a). If you can find one, I'd love to know.
 
timthereaper said:
From what I can tell, there is no good antiderivative of e^(st)/(Sqrt+a). If you can find one, I'd love to know.


Well, by using Mathematica, I got a result expressed in terms of error functions. So I guess I have to transform it into error functions. But, how?
 
Well, e^(st) can possibly be expressed as e^((Sqrt^2)*t), which when integrated with respect to s is similar to the error function. That might be the way to attack this.
 

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