# numerical integration - Numerical integration converging too slowly

I am trying to run the following code

l = N@Subdivide[-1, 1, 20][[;; -2]] + 1/20;
data = Table[NIntegrate[
1/2*r*
Log[r^2 + (x^2 + y^2) - 2 r*Sqrt[x^2 + y^2]*Cos[t - ArcTan[x, y]]]
, {t, 0, 2 Pi }, {r, 0, 1/Sqrt[4 Cos[t]^2 + 25 Sin[t]^2]},
PrecisionGoal -> 6], {x, l}, {y, l}];


and I am getting the error

NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.

The $$x$$ and $$y$$ are parameters for the integration. Is is possible to get the values of $$x$$ and $$y$$ for which I am getting the error?

• NIntegrate::slwcon is a warning, not an error. If the integral fails to converge, then it's a hint about what might be the problem. If the integral converges, then it can be ignored. (Of course, numerical methods might mess up, that is, give a result that is erroneous without giving any message about it.) Commented Jul 16, 2023 at 15:38

If you define

int[x_?NumericQ, y_?NumericQ] :=
NIntegrate[
1/2*r*Log[
r^2 + (x^2 + y^2) -
2 r*Sqrt[x^2 + y^2]*Cos[t - ArcTan[x, y]]], {t, 0, 2 Pi}, {r, 0,
1/Sqrt[4 Cos[t]^2 + 25 Sin[t]^2]},Method->"LocalAdaptive" ]


Plot3D shows a smooth surface without error message.

Plot3D[int[x, y], {x, -1, 1}, {y, -1, 1}]


Probably case x=0,y=0 causes the message in your solution grid!

Try

Table[{x, y, int[x, y]}, {x, Subdivide[-1, 1, 21]}, {y,Subdivide[-1, 1, 21]}]


which evaluates without error message