Problem with NIntegrate when WorkingPrecision is specified



I am trying to evaluate this integral numerically: $$ \int_0^{\infty } m \exp (-m) J_1(m){}^2 \, dm $$ Everything is OK when only the integration method is specified:

NIntegrate[-m Exp[-m] BesselJ[1, m]^2, {m, 0, Infinity}, Method -> "ClenshawCurtisRule"]

but when I specify the WorkingPrecision, the integral remains unevaluated:

NIntegrate[-m Exp[-m] BesselJ[1, m]^2, {m, 0, Infinity}, Method -> "ClenshawCurtisRule",
  WorkingPrecision -> 10]

What is wrong with this code?

I am using Mathematica v9.0.1


This bug is still present in version


Posted 2013-04-29T08:47:53.260

Reputation: 1 431

You can make it easier for others to check your code when you copy it straight from the Mathematica cell (copy as plain text) and paste it in your question with an indentation of 4 spaces. – Thies Heidecke – 2013-04-29T08:57:24.677

@ThiesHeidecke Codes are replaced with plain text. – M6299 – 2013-04-29T09:20:48.877



This is a bug. As a workaround for this specific integral you could use a symbolic solution:

Integrate[-m*Exp[-m]*BesselJ[1, m]^2, {m, 0, Infinity}]

(* (-3*EllipticE[-4] + 5*EllipticK[-4])/(5*Pi) *)


Posted 2013-04-29T08:47:53.260



"LevinRule" should work splendidly here, I think:

NIntegrate[-m Exp[-m] BesselJ[1, m]^2, {m, 0, Infinity}, 
           Method -> "LevinRule", WorkingPrecision -> 20]

ruebenko's answer has given a closed form for this particular definite integral. Personally, I prefer it when the parameters of the elliptic integrals are within $[0,1)$, so I apply the imaginary modulus transformations here to yield

N[(EllipticK[4/5] - 3 EllipticE[4/5])/(Sqrt[5] π), 20]

J. M.'s ennui

Posted 2013-04-29T08:47:53.260

Reputation: 115 520

yes of course, good point. Using a different method is certainly an option. – None – 2013-04-29T12:03:16.133

2This bug is apparently not present in version 7 so I added a version-9 tag; can you determine if this is in version 8? – Mr.Wizard – 2013-04-30T11:29:30.800

2@Mr. Wizard, Yes, it's busted in version 8. – J. M.'s ennui – 2013-04-30T11:37:03.543