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 
$$
\hat f(y)
=
\sqrt{2\pi a} (a^c) (b^c) (b^2+y^2)^{-c/2-1/4} BesselJ[c+1/2,a\sqrt{b^2+y^2}]
?
$$

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

Numerical computation suggests this is correct.

I need this formula for $c=1$.