       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!
>

In:= 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//InputForm= (E^((a^2*s^2)/2)*Sqrt[Pi/2]*Erfc[(a*s)/Sqrt])/s^2

Then the inverse Laplace transform is computed as below.

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

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

Daniel Lichtblau
Wolfram Research

```

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