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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