# a mysterious error in multiple symbolical integration

I am trying to do the following integration:

Ld = 2;
a1 = 0.3;
a2 = 1;

potd[R_, z_] =
-Integrate[(0.5/(2*a1)*Exp[-(Abs[z1]/a1)] +
0.5/(2*a2)*Exp[-(Abs[z1]/a2)])*
Integrate[
BesselK[0, a/Ld]*a*
ArcSin[(2*a)/(
Sqrt[(z - z1)^2 + (a + R)^2] +
Sqrt[(z - z1)^2 + (a - R)^2])], {a, 0, Infinity},
GenerateConditions -> False], {z1, -Infinity, Infinity},
GenerateConditions -> False];


then I evaluate the result for different pairs of R, z.

Mathematica complains that

NIntegrate::inumr: The integrand a ArcSin[(2 a)/(Sqrt[Power[<<2>>]+Power[<<2>>]]+Sqrt[Power[<<2>>]+Power[<<2>>]])] BesselK[0,a/2] has evaluated to non-numerical values for all sampling points in the region with boundaries {{∞,0}}. >>

The point is that there is no NIntegrate: what does it mean?

I upload the function from my notebook so you can see that I am not using a NIntegrate. • Have you tried it with a1 = 3/10? It might help to indicate specific R, z that give the message. Jul 30 '14 at 10:55
• @MichaelE2 that error shows up before I use any number. Jul 30 '14 at 11:33
• Under some circumstances, which I cannot explain in this case, Mathematica will use numerical methods in symbolic calculations. In this case, it seems to have to do with having approximate numeric coefficients, 0.5 and a1. If they are changed to exact numbers 1/2 and 1/3, the NIntegrate messages do not appear. (FWIW, M tries to evaluate NIntegrate[a ArcSin[(2 a)/(Sqrt[(a - R)^2 + (z - z1)^2] + Sqrt[(a + R)^2 + (z - z1)^2])] BesselK[0, a/2], {a, 0, ∞}, WorkingPrecision -> 30.9546, AccuracyGoal -> ∞, PrecisionGoal -> 20.9546], which is foolish because R and z are symbols.) Jul 30 '14 at 16:06
• @MichaelE2 thanks for the clarification ;) Jul 30 '14 at 17:47

I think I solved this problem.

The error message was due to the fact that Mathematica cannot perform the internal integration, so I split the two integrations and used NIntegrate instead of the symbolic integration:

p[z1_?NumericQ, R_?NumericQ, z_?NumericQ] :=
-NIntegrate[
BesselK[0, x/Ld]*x*
ArcSin[(2*x)/(
Sqrt[(z1)^2 + (x + 7.6)^2] + Sqrt[(z1)^2 + (x - 7.6)^2])], {x, 0,
Infinity}];
potd[R_?NumericQ, z_?NumericQ] :=
NIntegrate[(0.5/(2*a1)*Exp[-(Abs[z1]/a1)] +
0.5/(2*a2)*Exp[-(Abs[z1]/a2)])*p[z1, R, z], {z1, -Infinity,
Infinity}];
potd[7.6, 0]


with the result

-1.77324

Still: I don't undersand the previous error since I wasn't using numeric integration, but now it works...