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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?

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Have you tried with integral representations of the Bessel functions? – Raskolnikov Apr 4 '12 at 14:42
I tried but couldnt make progress after a certain point. – user16409 Apr 4 '12 at 23:35

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