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>