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MathGroup Archive 2006

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Re: Re: Fourier Transforms


Sseziwa Mukasa wrote:

>On Mar 2, 2006, at 6:48 AM, Ben C wrote:
>
>  
>
>>On the first of March I posted an appeal for help with some Fourier
>>transforms. Since then a couple of people have suggested I post the
>>actual transforms. I am trying to inverse Fourier transform the
>>functions
>>
>>p / (sqrt(1+p^2 + sqrt(1+p^2 ))   and   1/(sqrt(1+p^2 - sqrt(1+p^2 ))
>>
>>from p to x space.
>>
>>Any advice would again be extremely gratefully received,
>>    
>>
>
>You need to convert your expressions to proper Mathematica syntax,  
>sqrt(x) is Sqrt[x] in Mathematica.  Since you're looking for a  
>symbolic solution use InverseFourierTransform like so:
>
>In[3]:=
>InverseFourierTransform[p/Sqrt[1+p^2+Sqrt[1+p^2]],p,x]
>InverseFourierTransform[1/Sqrt[1+p^2-Sqrt[1+p^2]],p,x]
>Out[3]=
>0
>Out[4]=
>InverseFourierTransform[1/Sqrt[1+p^2-Sqrt[1+p^2]], p, x]
>
>The second expression returns itself because the integral cannot be  
>performed.  Perhaps your expression is only valid for p > 0?  You  
>also need to specify the convention for the Fourier transform you are  
>using, see the Help Browser for more information.
>
>Regards,
>
>Ssezi
>
>  
>
Just plotting your inputs, for arbitrary large values of p, they look 
similar to some simple functions (if one can call them that) whose FT 
can be easily ascertained. Here is my attempt anyway,
 >>
f1[p_] = p/Sqrt[1 + p^2 + Sqrt[1 + p^2]];
f2[p_] = 1/Sqrt[1 + p^2 - Sqrt[1 + p^2]];
Plot[{f1[p], Sign[p]}, {p, -1000, 1000}, PlotRange -> {-2, 2}];
Plot[f2[p], {p, -1000, 1000}];
InverseFourierTransform[Sign[p], p, x]
InverseFourierTransform[DiracDelta[p], p, x]

 >>
-Graphics-
-Graphics-
(I*Sqrt[2/Pi])/x
1/Sqrt[2*Pi]

I hope this is helpfull in some way

Pratik





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