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

[Stefan Jeglinski <jeglin@4pi.com>]

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