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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.

enter image description here

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After fixing several syntax errors one gets the following:

a = 0.1;
b = 0.01;
expr1 = Sqrt[x Exp[-x/a]] Sqrt[
   y Exp[-y/a]] (1 - Exp[1000 I (x + y)])/((x + y) (y - b))

(*  ((1 - E^(1000 I (x + y))) Sqrt[E^(-10. x) x] Sqrt[
 E^(-10. y) y])/((-0.01 + y) (x + y))  *)

Now we go to cylindrical coordinates:

expr2 = expr1 /. {x -> r*Cos[f], y -> r*Sin[f]} // 
  Simplify[#, {r >= 0, 0 < f < \[Pi]}] &

(*  -((1. (-1. + E^(1000 I r (Cos[f] + Sin[f]))) Sqrt[
  E^(-10. r Cos[f] - 10. r Sin[f]) r^2 Cos[f] Sin[f]])/(
 r (Cos[f] + Sin[f]) (-0.01 + r Sin[f])))   *)

Now let us integrate it over the angle first:

Integrate[f*r, {f, 0, \[Pi]}]

(*  (\[Pi]^2 r)/2   *)

It is clearly visible from here that the next integration (that is, over r from zero to infinity) will be infinite.

Thus your integral does not converge.

Have fun!

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  • $\begingroup$ Thank you. I fixed the errors in the code. $\endgroup$ – Farhad Dec 20 '16 at 12:57
  • $\begingroup$ Regarding your response, the integrand is kind of the dynamics of a damped harmonic oscillator. It essentially contains an exponential decay term and a sinusoidal term. The decay term makes sure that the function converges at infinity. This can be easily checked by setting 'a = 0.000001' and introducing "AccuracyGoal -> 10, MaxRecursion -> 300". This gives a number without any error. So it seems that the integral converges. $\endgroup$ – Farhad Dec 20 '16 at 12:59
  • $\begingroup$ For 'a = 0.1' obviously the convergence should be slow, but it finally converges. $\endgroup$ – Farhad Dec 20 '16 at 13:01
  • $\begingroup$ I inserted the plot of the integrand in the original post. Apart from the singularities, according to the plot, the integral clearly converges. $\endgroup$ – Farhad Dec 20 '16 at 13:32

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