Re: [Phys-l] fourier transform theorems for reciprocal functions?

[Brian Whatcott <betwys1@sbcglobal.net>]

Stefan Jeglinski wrote:
> Seems the list is in a summertime doldrum - not to be unexpected. Out 
> of curiosity then, I submit the following query:
>
> There exist general theorems associated with fourier transforms. For example:
>
> given:
> 	FourierTransform[f(x)] = F(k)
> then:
> 	FourierTransform[f(x-s)] = F(k)*Exp[-iks]
>
>
> These theorems range in complexity, and most/many are 
> straightforwardly derived. But I don't remember seeing any treatments 
> or discussion of something like this:
>
> given:
> 	FourierTransform[f(x)] = F(k)
> then:
> 	FourierTransform[1/f(x)] = ?
>
>
> I generally assume that the reason is because 1/f(x) is a totally 
> different function from f(x), and therefore there can be no general 
> expression; that is, it is a totally new FT and simply must be 
> calculated anew. Not to mention the difficulty of singularities if 
> f(x) is a "practical" function. But I'm just wondering if there is 
> *any* generalization that can be made, even a teeny tiny one, perhaps 
> for certain classes of f(x)?
>
> References appreciated, if they exist.
>
>
> Stefan Jeglinski
> _______________________________________________
> Forum for Physics Educators
> Phys-l@carnot.physics.buffalo.edu
> https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l
>
>   

I placed some pdfs from Alpha in an accessible place. I find the pdf
storage format tiresome, in that you need to download and open them,
but it’s the best I could do here. I may have mislabeled some files
with 'inverse' rather than 'reciprocal'.


FT t

http://www.datafilehost.com/download-9e41dda5.html

FT 1/ t

http://www.datafilehost.com/download-cf8bd522.html

FT t^2

http://www.datafilehost.com/download-a782c15e.html

FT 1/ ( t^2)

http://www.datafilehost.com/download-271c990d.html

FT (1 + t^2)

http://www.datafilehost.com/download-f2f66278.html

FT (1/(1 + t^2))

http://www.datafilehost.com/download-65811050.html

If these pdfs are accessible to you, the free datafile server
on which I placed them will also commend itself to you.

http://www.free-webhosts.com/image-hosting/DataFileHost/

(files are retained only while occasionally accessed)

Brian W
p.s. There don't seem to be FTs available for reciprocals of 
transcendental functions