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Mathematics Calculus Wolfram Language

Integrate a Bessel Function K0(a*x) without the constant 'a' ?

Romildo Junior

Posted 6 years ago

Thanks for reading ! I am having a problem with the numerical integration of the Bessel function y = BesselK0(a*x) Since my constant is too large (a = 6800) I am getting the large arguments aproximation for the Bessel K0 function, making the integration difficult. My question is: Is there some mathematical method or manipulation to remove the constant 'a' from the integration like y = BesselK0(x) to make me integrate it like small arguments and then insert the constant 'a' in it somehow after ? Thank you very much !

POSTED BY: Romildo Junior

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Romildo Junior

Posted 6 years ago

POSTED BY: Romildo Junior

Michael Rogers

Michael Rogers, Emory University

Posted 6 years ago

For this integral, the integrand has odd symmetry, so its principal value is zero.

POSTED BY: Michael Rogers

Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 6 years ago

It's seems a bug is in Integrate command and Yes you are right. Integrate[BesselK[0, (6632555543 Sqrt[xi^2])/135117312]/xi, {xi, -1, 1}, PrincipalValue -> True](* It should be Zero) Indefine integral: Integrate[BesselK[0, Sqrt[x^2]]/x, x](Cant find !) but: $$\int \frac{K_0\left(\sqrt{x^2}\right)}{x} \, dx=-\frac{1}{4} G_{1,3}^{3,0}\left(\frac{x^2}{4}\| \begin{array}{c} 1 \\ 0,0,0 \\ \end{array} \right)+C$$ With Cauchy Principal Value: Limit[((-Inactive[MeijerG][{{}, {1}}, {{0, 0, 0}, {}}, x^2/4]/4 // Activate) /. x -> (-e)) - ((-Inactive[MeijerG][{{}, {1}}, {{0, 0, 0}, {}}, x^2/ 4]/4 // Activate) /. x -> -1) + (((-Inactive[MeijerG][{{}, {1}}, {{0, 0, 0}, {}}, x^2/ 4]/4 // Activate) /. x -> 1) - ((-Inactive[MeijerG][{{}, {1}}, {{0, 0, 0}, {}}, x^2/ 4]/4 // Activate) /. x -> e)), e -> 0, Direction -> -1] ( 0 *)

It's seems a bug is in Integrate command and Yes you are right.

  Integrate[BesselK[0, (6632555543 Sqrt[xi^2])/135117312]/xi, {xi, -1, 1}, PrincipalValue -> True](* It should be Zero*)

Indefine integral:

Integrate[BesselK[0, Sqrt[x^2]]/x, x](*Cant find !*)

but:

$$\int \frac{K_0\left(\sqrt{x^2}\right)}{x} \, dx=-\frac{1}{4} G_{1,3}^{3,0}\left(\frac{x^2}{4}| \begin{array}{c} 1 \\ 0,0,0 \\ \end{array} \right)+C$$

With Cauchy Principal Value:

  Limit[((-Inactive[MeijerG][{{}, {1}}, {{0, 0, 0}, {}}, x^2/4]/4 // 
        Activate) /. 
      x -> (-e)) - ((-Inactive[MeijerG][{{}, {1}}, {{0, 0, 0}, {}}, x^2/
          4]/4 // Activate) /. 
      x -> -1) + (((-Inactive[MeijerG][{{}, {1}}, {{0, 0, 0}, {}}, x^2/
            4]/4 // Activate) /. 
        x -> 1) - ((-Inactive[MeijerG][{{}, {1}}, {{0, 0, 0}, {}}, x^2/
            4]/4 // Activate) /. x -> e)), e -> 0, Direction -> -1]

    (*  0  *)

POSTED BY: Mariusz Iwaniuk

Romildo Junior

Posted 6 years ago

This is a singular integral, maybe mathematica is having some problems integrating it ?

POSTED BY: Romildo Junior

Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 6 years ago

func = Rationalize[(Sqrt[(0. + 0.0078125 (1 - xi) + 0.0078125 xi)^2] (-0.0078125 + 0.0078125 (1 - xi) (1 + xi) + 0.0078125 xi (1 + xi)) BesselK[ 0, 6283.185307179586* Sqrt[0. + (-0.0078125 + 0.0078125 (1 - xi) (1 + xi) + 0.0078125 xi (1 + xi))^2]])/(0. + (-0.0078125 + 0.0078125 (1 - xi) (1 + xi) + 0.0078125 xi (1 + xi))^2), 0] // FullSimplify; NIntegrate[func, {xi, -1, 0, 1}, Method -> PrincipalValue] (* 2.5243510^-29 ) With the Cauchy principal value integral is finite and equal to Zero.

 func = Rationalize[(Sqrt[(0. + 0.0078125 (1 - xi) + 
             0.0078125 xi)^2] (-0.0078125 + 
           0.0078125 (1 - xi) (1 + xi) + 0.0078125 xi (1 + xi)) BesselK[
          0, 6283.185307179586*
           Sqrt[0. + (-0.0078125 + 0.0078125 (1 - xi) (1 + xi) + 
                0.0078125 xi (1 + xi))^2]])/(0. + (-0.0078125 + 
            0.0078125 (1 - xi) (1 + xi) + 0.0078125 xi (1 + xi))^2), 
      0] // FullSimplify;
  NIntegrate[func, {xi, -1, 0, 1}, Method -> PrincipalValue]
  (* 2.52435*10^-29 *)

With the Cauchy principal value integral is finite and equal to Zero.

POSTED BY: Mariusz Iwaniuk

Romildo Junior

Posted 6 years ago

Thank you very much for playing with the integral, yes this one is divergent !! But this is at one node of the problem, I am also dealing with integrals that are not divergent like this one: Integrate[(Sqrt[(0. + 0.0078125(1 - xi) + 0.0078125xi)^2](-0.0078125 + 0.0078125(1 - xi)(1 + xi) + 0.0078125xi(1 + xi)) BesselK[0, 6283.185307179586* Sqrt[0. + (-0.0078125 + 0.0078125(1 - xi)(1 + xi) + 0.0078125xi(1 + xi))^2]])/ (0. + (-0.0078125 + 0.0078125(1 - xi)(1 + xi) + 0.0078125xi(1 + xi))^2), {xi, -1, 1}] If the correct numerical integration is used, this integral is given = 0. So if the constant is used 6283.185307179586, one need to solve as a cauchy principal value integral, if the constant is small one could solve as a weakly singular integral. I would love to remove that constant from the Bessel Term and then add it somehow after the integration. But Thank you very much for your answer !

POSTED BY: Romildo Junior

Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 6 years ago

func = Rationalize[(Sqrt[(0. + 0.0078125 (1 - xi) + 0.0078125 xi)^2] (-0.015625 + 0.0078125 (1 - xi) (1 + xi) + 0.0078125 xi (1 + xi)) BesselK[0, 6283.185307179586* Sqrt[0. + (-0.015625 + 0.0078125 (1 - xi) (1 + xi) + 0.0078125 xi (1 + xi))^2]])/(0. + (-0.015625 + 0.0078125 (1 - xi) (1 + xi) + 0.0078125 xi (1 + xi))^2), 0] // FullSimplify Integrate[func, {xi, -1, 1}](Integral is divergent) Integrate[BesselK[0, 2 Sqrt[(-1 + xi)^2]]/(-1 + xi), {xi, -1, 1}] (Simplified version.Integral is divergent) Removing this: '6283.185307179586' large number will not help you at all.

  func = Rationalize[(Sqrt[(0. + 0.0078125 (1 - xi) + 
            0.0078125 xi)^2] (-0.015625 + 0.0078125 (1 - xi) (1 + xi) + 
          0.0078125 xi (1 + xi)) BesselK[0, 
         6283.185307179586*
          Sqrt[0. + (-0.015625 + 0.0078125 (1 - xi) (1 + xi) + 
               0.0078125 xi (1 + xi))^2]])/(0. + (-0.015625 + 
           0.0078125 (1 - xi) (1 + xi) + 0.0078125 xi (1 + xi))^2), 0] //
     FullSimplify
  Integrate[func, {xi, -1, 1}](*Integral is divergent*)

  Integrate[BesselK[0, 2 Sqrt[(-1 + xi)^2]]/(-1 + xi), {xi, -1, 1}]
  (*Simplified version.Integral is divergent*)

Removing this: '6283.185307179586' large number will not help you at all.

POSTED BY: Mariusz Iwaniuk