       Re: Bug in LaplaceTransform?

• To: mathgroup at smc.vnet.net
• Subject: [mg97740] Re: [mg97693] Bug in LaplaceTransform?
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Fri, 20 Mar 2009 02:39:49 -0500 (EST)
• References: <200903190709.CAA23011@smc.vnet.net>

```Wieland Brendel wrote:
> how can it happen that Mathematica throws out -i for the Laplace
> transform of one,
>
> LaplaceTransform[1, t, I] = -i?
>
> After all the Laplace transformation (with s = I) is defined as
>
> Integrate[Exp[-I t], {t, 0, Infinity}]
>
> and should be undefined. Am I wrong or is this a bug?
>
> Thanks for an answer!
> Wieland

This result is correct and (by now) classical. Quoting from "Generalized
Functions: Theory and Technique" by Ram Kanwal: "The Laplace transform
of the Heaviside function is...1/s"

Some ways to derive this involve regularizing and computing in a
limiting sense. Could do as

In:= Integrate[Exp[-I*s*t], {t,0,Infinity}, Assumptions->Im[s]<0]
Out= -I/s

Now let s approach 1 from below (in the complex plane).

Or as

In:= InputForm[l2 = Integrate[Exp[-I*t]*t^a,
{t,0,Infinity}, Assumptions->-1/1000<a<0]]
Out//InputForm= ((-I)*Gamma[1 + a])/E^((I/2)*a*Pi)

In:= Limit[l2,a->0]
Out= -I

Daniel Lichtblau
Wolfram Research

```

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