how can we prove that the Fourier transform of the function

(a^2-x^2)^(c/2) * BesselJ[c,b*Sqrt[a^2-x^2]] for x^2 < a^2 (and 0 elsewhere)

is

Sqrt[2Pia] * (a^c) * (b^c) * (b^2+y^2)^(-c/2-1/4) * BesselJ[c+1/2,aSqrt[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.

Thanks for any help

Pavel