Re: Shifted & scaled Heaviside and FT
- To: mathgroup at smc.vnet.net
- Subject: [mg122575] Re: Shifted & scaled Heaviside and FT
- From: "Dr. Wolfgang Hintze" <weh at snafu.de>
- Date: Wed, 2 Nov 2011 06:20:17 -0500 (EST)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
- References: <j8omuq$lfg$1@smc.vnet.net>
"Anael" <anael.guilmo at gmail.com> schrieb im Newsbeitrag news:j8omuq$lfg$1 at smc.vnet.net... > Hi all, > > I am having trouble with computing the Fourier Transform of a shifted > AND scaled Heaviside function. > > Basically I can get: > FourierTransform[HeavisidePi[(x + a)], x, \[Xi]] = > (E^(-I a \[Xi]) Sinc[\[Xi]/2])/Sqrt[2 \[Pi]] > > or > FourierTransform[HeavisidePi[(x/b)], x, \[Xi]] = > (Abs[b] Sinc[(b \[Xi])/2])/Sqrt[2 \[Pi]] > > but when I combine [(x + a)/ b] i get nothing! > Any clue?? > > Thank you! > Sorry, I know this might not help you, but I have no difficulty (after having defined in my older version: HeavisidePi[x_] := UnitStep[1/2 + x] - UnitStep[x - 1/2]) (* 1 shifted only *) In[67]:= FourierTransform[HeavisidePi[x + a], x, \[Xi]] Out[66]= (Sqrt[2/Pi]*Sin[\[Xi]/2])/(E^(I*a*\[Xi])*\[Xi]) (* 2 shifted and scaled *) In[67]:= FourierTransform[HeavisidePi[(x + a)/b], x, \[Xi]] Out[67]= (1/(Sqrt[2*Pi]*\[Xi]))*(I*((Cos[a*\[Xi] - (b*\[Xi])/2] + I*Sin[\[Xi]/(2*Sqrt[1/(-2*a + b)^2])])* (-1 + UnitStep[-b]) - (Cos[a*\[Xi] + (b*\[Xi])/2] + I*Sin[\[Xi]/(2*Sqrt[1/(2*a + b)^2])])* (-1 + UnitStep[-b]) + (-1 + UnitStep[b])/E^((1/2)*I*(2*a + b)*\[Xi]) - E^((-I)*a*\[Xi] + (I*b*\[Xi])/2)*(-1 + UnitStep[b]))) In[69]:= Simplify[%, b > 0] Out[69]= (Sqrt[2/Pi]*Sin[(b*\[Xi])/2])/\[Xi] - I*b*Sqrt[2*Pi]*DiracDelta[b*\[Xi]]*Sin[(b*\[Xi])/2] (* 3 scaled only *) In[70]:= FourierTransform[HeavisidePi[x/b], x, \[Xi]] Out[70]= (Sqrt[2/Pi]*Sin[(b*\[Xi])/2])/(\[Xi]*Sign[b]) - (I*b*Sqrt[2*Pi]*DiracDelta[b*\[Xi]]*Sin[(b*\[Xi])/2])/Sign[b] In[71]:= Simplify[%, b > 0] Out[71]= (Sqrt[2/Pi]*Sin[(b*\[Xi])/2])/\[Xi] - I*b*Sqrt[2*Pi]*DiracDelta[b*\[Xi]]*Sin[(b*\[Xi])/2] Best regards, Wolfgang