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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:
> Dear reader,
> 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[9]:= Integrate[Exp[-I*s*t], {t,0,Infinity}, Assumptions->Im[s]<0]
Out[9]= -I/s

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

Or as

In[13]:= InputForm[l2 = Integrate[Exp[-I*t]*t^a,
   {t,0,Infinity}, Assumptions->-1/1000<a<0]]
Out[13]//InputForm= ((-I)*Gamma[1 + a])/E^((I/2)*a*Pi)

In[14]:= Limit[l2,a->0]
Out[14]= -I

Daniel Lichtblau
Wolfram Research

• References:
  • Bug in LaplaceTransform?
    • From: Wieland Brendel <wielandbrendel@gmx.net>