How can we prove that the Fourier transform of the function $$ f(x) = \begin{cases} (a^2-x^2)^{c/2} BesselJ[c,b\sqrt{a^2-x^2}] & \text{for }x^2 < a^2\\ 0 & \text{otherwise} \end{cases} $$ is $$ \hat f(y) = \sqrt{2\pi a} \times (a^c) \times (b^c) \times (b^2+y^2)^{-c/2-1/4} \times BesselJ[c+1/2,a\sqrt{b^2+y^2}] ? $$

This Fourier transform pair is given in the book Formeln und Satze fur die speziellen Funktionen der mathematischer Physik (Julius Springer, Berlin, 1943) p. 119.

http://www.pokyrek.cz/Nijboer/Magnus_Hettinger.pdf

Numerically computation suggests this is correct.

I need this formula for $c=1$.