This is related to my previous post, which didn't resolve. I want to calculate the principal value of the following two-dimensional integral
$$ \int_{0}^{\infty}dx\int_{0}^{\infty}dy\sqrt{xe^{-10x}}\sqrt{ye^{-10y}}\frac{1-e^{1000\imath(x+y)}}{(x+y)(y-0.01)} $$
The mathematica code is
a=0.1;
b=0.01;
NIntegrate[Sqrt[x E^{-x/a}]Sqrt[y E^{-y/a}](1-E^{1000I(x+y)})/((x+y)(y-b)),{x,0,∞},{y,0,∞}]
But it gives the slwcon error:
Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small
I searched all over the internet. I tried different numerical integration methods (GlobalAdaptive, LocalAdaptive, ...) with different options (AccuracyGoal,PrecisionGoal, ...) but none of them get rid of the error. The integral for very small values of a converges and gives the result without error. This means that the integral is convergent. But I need the result for the values specified in the code above. How should I resolve the error?
Edit: I plotted the integrand in terms of its variables x, y. As you can see the integrand converges, although it has a singularity point y=0.01 and a singularity line x+y=0.