Fourier Transform of the following function: $\xi(\alpha,\tilde{\alpha},\beta)=\frac{J_{1}\left(\varrho_{0}k_{0}n\sqrt{\sin^{2}\alpha+\sin^{2}\tilde{\alpha}-2\sin\alpha\,\sin\tilde{\alpha}\cos\beta}\right)}{\sqrt{\sin^{2}\alpha+\sin^{2}\tilde{\alpha}-2\sin\alpha\,\sin\tilde{\alpha}\cos\beta}}$ , i.e to find $\hat{\xi}(\alpha,\tilde{\alpha},m)=\frac{1}{2\pi}\int_{0}^{2\pi}\xi(\alpha,\tilde{\alpha},\beta)\exp(-im\beta)d\beta$
I tried using addition theorem of Bessel functions and I get quite cumbersome expressions. Any help in getting it from standard integrals?