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Re: Fourier Transform

• To: mathgroup at smc.vnet.net
• Subject: [mg124266] Re: Fourier Transform
• From: "Oleksandr Rasputinov" <oleksandr_rasputinov at hmamail.com>
• Date: Sat, 14 Jan 2012 17:16:39 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jercl1$1d8$1@smc.vnet.net>

On Sat, 14 Jan 2012 07:59:29 -0000, <jstorch at earthlink.net> wrote:

> Mathematica hangs when trying to evaluate the simple Fourier Transform
> of H(a-|t|).
>
> FourierTransform[HeavisideTheta[a-abs[t]],t,w,Assumptions->a>0]
>
> (The answer should be Sqrt(2/Pi) (Sin[w a])/w  )
>

Typo.

In[1]:=
FourierTransform[
  HeavisideTheta[a - Abs[t]], t, w,
  Assumptions -> a > 0
]

Out[1]=
-((I E^(-I a w) (-1 + E^(2 I a w)))/(Sqrt[2 \[Pi]] w))

In[2]:=
% // Simplify

Out[2]=
(Sqrt[2/Pi] Sin[a w])/w

By the way, even with undefined abs[t], Mathematica doesn't hang. You just  
didn't give it long enough to figure out that it can't produce a useful  
result for the FT of H(a-f(t)) where f is an arbitrary function. In  
version 7 this takes about 30 seconds and in version 8 about two minutes  
on my computer.