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Ramanujan's Integral


 int_(-infty)^infty(J_(mu+xi)(x))/(x^(mu+xi))(J_(nu-xi)(y))/(y^(nu-xi))e^(itxi)dxi 
 =[(2cos(1/2t))/(x^2e^(-it/2)+y^2e^(it/2))]^((mu+nu)/2)
 ×J_(mu+nu)[sqrt(2cos(1/2t)(x^2e^(-it/2)+y^2e^(it/2)))]e^(it(nu-mu)/2),

where J_n(z) is a Bessel function of the first kind.


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References

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

Referenced on Wolfram|Alpha

Ramanujan's Integral

Cite this as:

Weisstein, Eric W. "Ramanujan's Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RamanujansIntegral.html

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