Ramanujan's integral is given by
![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),](/images/equations/RamanujansIntegral/NumberedEquation1.svg) |
where 
is a Bessel function of the first kind.
Ramanujan's Integral
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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 IntegralCite this as:
Weisstein, Eric W. "Ramanujan's Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RamanujansIntegral.html