Re: Why the FourierTransform gives two different answers?

*To*: mathgroup at smc.vnet.net*Subject*: [mg125134] Re: Why the FourierTransform gives two different answers?*From*: "Nasser M. Abbasi" <nma at 12000.org>*Date*: Thu, 23 Feb 2012 05:48:07 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <jhvupg$893$1@smc.vnet.net>*Reply-to*: nma at 12000.org

On 2/21/2012 5:22 AM, Ð?Ð»ÐµÐºÑ?ÐµÐ¹ wrote: > Why the FourierTransform gives two different answers? > > In[1] FourierTransform[ (t - 5.0)^2*Exp[-(t - 5.0)^2 ], t, w] > > In[2] FourierTransform[ (t - 5)^2* Exp[-(t - 5)^2 ], t, w] > It seems to be in integration part. In this below I used the text book definition of the F.T. itself, and instead of integrating to infinity, integrate to 'k', then take the limit as k->Infinity of the result of the integration, and I get the same result of both the symbolic and the numerical version. So, the problem is with the limit to infinity with the numerical version inside the integration code. -----------------numerical ------------------ ClearAll[t, w, k] f = (t - 5.0)^2*Exp[-(t - 5.0)^2]; res = Integrate[f*Exp[-I w t], {t, -k, k}]; res = Limit[res, k -> Infinity]; Plot[{Re[res], Im[res]}, {w, -5, 5}] ------------------------- ------------ symbolic ----------- ClearAll[t, w, k] f = (t - 5)^2*Exp[-(t - 5)^2]; res = Integrate[f*Exp[-I w t], {t, -k, k}]; res = Limit[res, k -> Infinity]; Plot[{Re[res], Im[res]}, {w, -5, 5}] ----------------------------- same result, as what one would expect. If I change the numerical integration above to be Integrate[f*Exp[-I w t], {t, -Infinity, Infinity}] Then it become zero again. So, I think it is a bug in the integration with the limit to infinity for this integrand. --Nasser