I have been trying to perform a Numerical Integration using the NIntegrate function. The problem I am facing is that on using the Integrand as a Bessel Function of Imaginary component, the result just fails to converge and the answer just blows up to some large number. None of the methods seem to give the correct result. The sample Integrand which I am trying to Integrate is:

fun[b1_, b2_, x1_, x2_] := 
 BesselK[0, Sqrt[-x1*b1]]*BesselI[0, Sqrt[-x2*b2]]

 fun[b1, b2, x1, x2], {b1, 0, Infinity}, {b2, 0, Infinity}, {x1, 0, 
  1}, {x2, 0, 1}, Method -> "AdaptiveMonteCarlo", AccuracyGoal -> 10]


During evaluation of In[113]:= 

Out[113]= -8.1181*10^8 - 7.19087*10^9 I    

This is the complete integrand I am trying to work on. Please do check for this integrand.

ha[b1_, b2_, x1_, 
      x2_] := ((b1 - b2 + 2)*
         BesselI[0, Sqrt[\[Beta]a[x1, x2, rJ\[Psi], rb, mB]]*b1]*
          Sqrt[\[Beta]a[x1, x2, rJ\[Psi], rb, mB]]*b2] + \[Theta][
          b2 - b1]
         BesselI[0, Sqrt[\[Beta]a[x1, x2, rJ\[Psi], rb, mB]]*b2]*
         BesselK[0, Sqrt[\[Beta]a[x1, x2, rJ\[Psi], rb, mB]]*b1])*
      BesselK[0, Sqrt[\[Alpha][x1, x2, rJ\[Psi], mB]*b1]]

The functions in the integrand are:

   \[Alpha][x1_, x2_, rJ\[Psi]_, 
      mB_] := -((x1 - x2)*(x1 - x2*rJ\[Psi]^2))*mB^2
    \[Beta]a[x1_, x2_, rJ\[Psi]_, rb_, 
      mB_] := -((1 - x2)*(1 - x2*rJ\[Psi]^2) - rb^2)*mB^2

mB, rJpsi,rb, etc are constant values.

  • $\begingroup$ MonteCarlo methods badly handle divergent improper multidimensional integrals. $\endgroup$
    – user64494
    Feb 25, 2021 at 17:24
  • $\begingroup$ What do you suggest to use. I am currently stuck in this part $\endgroup$
    – Illlusion5
    Feb 25, 2021 at 17:36
  • $\begingroup$ Illlusion5 (@ does not work.) : I don't know good numeric methods for improper multidimensional integrals. $\endgroup$
    – user64494
    Feb 25, 2021 at 18:13

1 Answer 1


The integral under consideration diverges. The one splits into the product of two double improper integrals. One of these is

Integrate[BesselK[0, Sqrt[-x1*b1]], {b1, 0, Infinity}, {x1, 0, 1}]

which performs

"Integral of -(2/b1)+(2\I\BesselK[1,I\Sqrt[b1]])/Sqrt[b1] does not converge on {0,[Infinity]}"


$\int_0^{\infty } \frac{-2+2 i \sqrt{\text{b1}} K_1\left(i \sqrt{\text{b1}}\right)}{\text{b1}} \, d\text{b1}$

  • 1
    $\begingroup$ Yes! I am getting the same result with Integrate. But the problem is with NIntegrate. The result should be some finite value. Actually this is a oversimplified version of the integrand I am actually working on. $\endgroup$
    – Illlusion5
    Feb 25, 2021 at 14:22
  • $\begingroup$ "Mathematics is like a mill: if you fill it with wheat grains, you will get flour, but if you fill it with bran, you will get bran" Andrew Huxley $\endgroup$
    – user64494
    Feb 25, 2021 at 15:32
  • $\begingroup$ @Illlusion5: Can you present a more complicated version of your code? In other case this is an empty talk. $\endgroup$
    – user64494
    Feb 25, 2021 at 16:36
  • $\begingroup$ Please do check for the following function. $\endgroup$
    – Illlusion5
    Feb 25, 2021 at 17:31

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