hankel transform (fourier-bessel of zero order)

[ElDonAntonio]

Is there anyone who would know by any chance the steps to find the hankel transform (also called fourier-bessel of zero order) of a function? I know the equation of the Hankel transform, but I just don't understand how to simplify it.
I'm actually trying to find the hankel transform of a dirac's delta. I found the answer on the web, but I'd like to know how to get to it myself.

 

[Fred]

All those transforms are of the form

G(x) = integral (0.. infinity) [f(t) K (x, t) ] dt

G(x) is the integral transform of f(t) and K(x,t) is the so called kernel.

for Fourier analysis the kernel is exp (i x t) (and a -infinity..infinity i think in the integral), for laplace transforms exp (-xt)

For Hankel transforms, It should be J n (xt) where J n is some bessel function of order n. Don't remember the exact form off the top of my head, but you can look it up.

The vanilla delta function is defined by f(t) = int (-infinity...infinity) delta (x-x0) dt = f(x0)

Either way, are you sure you're trying to take the hankel transform of the delta function, or rather trying to derive the delta function in terms of bessel functions. Because the former is easy, its just going to output the answer

J n (X0)

 

[ElDonAntonio]

Thanks Fred! actually I found the answer right after posting, realising everything in the integral was a constant except the delta function, which evaluates to 1.

I now have another problem. How do I calculate the Hankel transform of a rectangle? this is trickier, I can't dismiss anything in the integral. The answer seems to output a Bessel function of order 1, but I don't see the link.

 

[ElDonAntonio]

I think I just understood how to do it for the rectangle. The rectangle function will simply change the bounds of integration. And the integral of J0 ends up being a term in J1 (can easily be shown if you integrate the Bessel series of 0 order).