# Solving numerically a double integral that converges slowly?

Posted 2 months ago
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 I'm trying to solve numerically the following double integral in Mathematica, which I know, a priori, has a solution for values of $mh \in[0.4,2.0]$ and tmer = 0.017, tmel = 0.044.  ID[mh_] := 1/(2*Pi^3)* NIntegrate[(mh*tmer^(1/ 2)*(tmer^(1/2)*Sqrt[w^2 - 1]* Sqrt[(u*mh - w*tmer^(1/2))^2 - tmel] + 1/2*(mh^2 - tmer - tmel) + (w*tmer^(1/2))^2 - u*w*mh*tmer^(1/2)))* Exp[u*mh]/((Exp[u*mh] - 1)*(Exp[w*tmer^(1/2)] + 1)*(Exp[u*mh - w*tmer^(1/2)] + 1)), {u, 1, +\[Infinity]}, {w, 1, +\[Infinity]}] If you run, for instance, ID[0.4] you'll see that the numerical integration is converging too slowly, I suppose due to the Exp[u*mh] term in the numerator. Furthemore, it returns a complex number, thanks to Sqrt[(u*mh - w*tmer^(1/2))^2 - tmel]. However, I want to impose that (u*mh - w*tmer^(1/2))^2 >= tmel. If done correctly, I would expect ID[0.4] to output 0.000025385 or close to it. Is there a way to deal with the mentioned slow convergence and impose the above condition?
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Posted 2 months ago
 Try replacing Sqrt[...] with Surd[..., 2].