Ramanujan's Integral

$\int_{-\infty}^\infty {J_{\mu+\xi}(x)\over x^{\mu+\xi}} {J_{\nu-\xi}(y)\over y^{\nu-\xi}} e^{it\xi}\,d\xi$

$= \left[{2\cos\left({{1\over 2}t}\right)\over x^2e^{-it/2}+y^2e^{it/2}}\right]... ...tyle{1\over 2}}t}\right)(x^2e^{-it/2}+y^2e^{it/2})}\,}\right]e^{it(\nu-\mu)/2},$

where is a Bessel Function of the First Kind.

References

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.

$\int_{-\infty}^\infty {J_{\mu+\xi}(x)\over x^{\mu+\xi}} {J_{\nu-\xi}(y)\over y^{\nu-\xi}} e^{it\xi}\,d\xi$
$= \left[{2\cos\left({{1\over 2}t}\right)\over x^2e^{-it/2}+y^2e^{it/2}}\right]... ...tyle{1\over 2}}t}\right)(x^2e^{-it/2}+y^2e^{it/2})}\,}\right]e^{it(\nu-\mu)/2},$