I am working on using Fast Sine Transform to calculate the Hankel transform and its inverse with the Bessel Function of order 1/2. $$F(k)=\frac{4\pi}{k}\int_{0}^{\infty}\sin(kr)rf(r)dr$$
$$f(r) = \frac{1}{2\pi^{2}r}\int_{0}^{\infty}\sin(kr)kF(k)dk$$
It looks like the inverse transform diverge at the point $$\lim_{r\to0} f(r) = \frac{1}{2\pi^{2}r}\int_{0}^{\infty}\sin(kr)kF(k)dk$$ Is this true? How do people calculate radial distribution function at $r=0$ from X-Ray diffraction data?