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Re: does anybody know how to find the inverse Laplace transform of this wierd thing?

• To: mathgroup at smc.vnet.net
• Subject: [mg64127] Re: [mg64100] does anybody know how to find the inverse Laplace transform of this wierd thing?
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Thu, 2 Feb 2006 00:05:37 -0500 (EST)
• References: <200602010934.EAA23043@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

gino wrote:
> Want to find the inverse Laplace transform of the following term:
> 
> H(s)=1/s^2*exp(s^2*a^2/2)*integrate(exp(-u^2/2), u from s*a to +infinity)
> 
> How to do that?
> 
> ------------------------------
> 
> Making relaxation to the problem, if I have to find only certain sampled 
> values of the inverse Laplace transform of H(s), let's denote it as h(t),
> 
> I just need to find h(1), h(2), h(3), etc.
> 
> Is there a short cut for it?
> 
> Thanks a lot! 
> 

We start with your input.

In[9]:= InputForm[hH[s_] =
         1/s^2*Exp[s^2*a^2/2]*Integrate[Exp[-u^2/2],
         {u,s*a,Infinity}, Assumptions->Element[a,Reals]]]
Out[9]//InputForm= (E^((a^2*s^2)/2)*Sqrt[Pi/2]*Erfc[(a*s)/Sqrt[2]])/s^2

Then the inverse Laplace transform is computed as below.

In[11]:= InputForm[InverseLaplaceTransform[hH[s], s, t,
          Assumptions->Element[a,Reals]]]
Out[11]//InputForm=
Sqrt[Pi/2]*(a*(-1 + E^(-t^2/(2*a^2)))*Sqrt[2/Pi] + t*Erf[t/(Sqrt[2]*a)])

You will need to decide for yourself whether this is what you had in mind.

Daniel Lichtblau
Wolfram Research

• References:
  • does anybody know how to find the inverse Laplace transform of this wierd thing?
    • From: "gino" <loseminds@hotmail.com>