How can I solve the problem with Fourier transforms in critical phenomena?

[enrikofermi]

Hi all.
I'm learning something about critical phenomena and I have one problem.

I'm bad with Fourier transforms so I don't know how from 7.37 we have 7.38.
I have tryed everything I knew, but fruitless. I have attached picture of my problem.

Does anybody has any idea how I can solve this?

 

[DrDu]

Multiply 7.37 with the denominator of the RHS and then Fourier transform using that a product in Fourier space becomes a convolution in direct space (and vice versa), i.e. the integral on the RHS of eq. 7.38.

https://en.wikipedia.org/wiki/Convolution_theorem

 

[enrikofermi]

Thanks!
Is the Fourier transform of C (with hat) just C without hat, taking into account deffinition of C?
What make me confused are the arguments of functions in 7.38. Why r-r', and so on...?

 

[enrikofermi]

Anybody? To help with this manybody? :)

 

[DrDu]

That C and $\hat{C}$ are Fourier transforms of each other is clearly stated in the article.
r1-r2 is as valid a variable as e.g. x or y. In a homogeneous system, translation invariant quantities cannot depend on both r1 and r2 but via their difference r1-r2. I suppose this has been discussed in your text previously

 

[enrikofermi]

Yes, I know, it was discussed earlier. But that what confused me was arguments of those functions under integral. Why are they different? And why they differ from those arguments of function outside of integral?

 

[DrDu]

I think you first have to convince yourself that the integral represents a convolution.

 

[enrikofermi]

I have used convolution theorem, but I didn't get exactly same solution as author. Look how I calculated that. Arguments of my functions are considerably different. And that is what confuses me...

 

[DrDu]

Rename your variables:
r->r-r'
And
r''->r''-r'

 

[enrikofermi]

Thank you very much Doc!