What is the Fourier transform of this function ?

[hiyok]

Hi, I have problems finding out the Fourier transform of the following function,

1/\sqrt{q^2 + m^2}, where m\neq 0 denotes a parameter.

It seems easy, but I don't know how. Could anybody show me how to do it ?

Thanks in advance.

hiyok

 

[Simon Bridge]

$$f(q)=\frac{1}{\sqrt{q^2+m^2}}\\ \mathcal{F}(p)=\int_{-\infty}^\infty \frac{e^{-2\pi iqp}}{\sqrt{q^2+m^2}}\;\text{d}q$$ ... this correct?
i.e. you want the forward Fourier transform...

Please show your best attempt.

 

[hiyok]

Yes, that is exactly what I meant.

I tried to make a contour and evaluate the residue around the pole q=im. But the order of this pole is not integer. I don't know how to proceed.

Thanks

 

[Simon Bridge]

Looks nasty - I think the solution may have Bessel functions.
http://www.wolframalpha.com/input/?...f3=p&f=FourierTransformCalculator.variable2_p
Where did this come from?

 

[hiyok]

Thanks a lot for your useful message. I'll look into your link.

I met this integral when trying to find out the Green's function for a graphene ribbon.