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MathGroup Archive 2002

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Re: Re: A Bessel integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg36887] Re: Re: A Bessel integral
  • From: Vladimir Bondarenko <vvb at mail.strace.net>
  • Date: Tue, 1 Oct 2002 04:45:08 -0400 (EDT)
  • Reply-to: Vladimir Bondarenko <vvb at mail.strace.net>
  • Sender: owner-wri-mathgroup at wolfram.com

On Sun, 29 Sep 2002 09:35:41, in the message Re: A Bessel integral,
Tom Burton <tburton at brahea.com> wrote:

TB> On 9/28/02 11:54 PM, in article an68bt$s7$1 at smc.vnet.net, "Vladimir
TB> Bondarenko" <vvb at mail.strace.net> wrote:

VB>> The expression for W[m_,n_] returned by Mathematica is wrong.
VB>>
VB>> To prove, just substitute m = n = 0 which is exactly what you had done
VB>>
VB>> and observe that the output you had had
VB>>
VB>> W[0,0]=-(2 EulerGamma + Log[4] + 4 PolyGamma[0, 1/2])/(2 Pi)
VB>>
VB>> = 0.84564
VB>>
VB>> was incorrect. The correct answer is 1/2.
                    ^^^^^^^^^^^^^^^^^^^^^^^^^^

TB> W[0,0]diverges. Mathematica gets that wrong.

Thank you for your correction!

(That my terrible bug shows how it is dangerous to do several
things at a time 8-(  From now on, I promise to reread my
posting to the MathGroup before sending them ;-)

Why sure, you are right, the integral

     Integrate[BesselJ[0, z]^2, {z, 0, Infinity}]

diverges because the integrand is bounded everywhere
over the integration region and decays at z -> Infinity
as Cos[Pi/4 - z]^2/z + o(z), that is as

     In[1] := Expand[TrigExpand[Cos[Pi/4 - z]^2/z]] // InputForm
     Out[1] = 1/(2*z) + (Cos[z]*Sin[z])/z

which means that the integral

      Integrate[BesselJ[0, z]^2, {z, 0, x}]

diverges logarithmically in x.


By the way, the main term of

      Expand[Normal[Series[BesselJ[0, z], {z, Infinity, 1}]]^2]

is (2*Cos[Pi/4 - z]^2)/(Pi*z) which conveys the suggestion that
we should try it, too.

This reveals us another integral which Mathematica 4.1 fails to calculate

    In[1] := Integrate[Cos[Pi/4 - z]^2/z, {z, 1, Infinity}] // N
    Out[1]=  -0.0173083

    In[2] := $Version
    Out[2]=  "4.1 for Microsoft Windows (November 2, 2000)"

but Mathematica 4.2 handles correctly

    In[1] := Integrate[Cos[Pi/4 - z]^2/z, {z, 1, Infinity}]
    Out[1] = Integrate::"idiv": "Integral of"... "does not converge on {1, Infinity)."

    In[2] := $Version
    Out[2]=  "4.2 for Microsoft Windows (February 28, 2002)"


Even simpler,

    In[1] := Integrate[Cos[z]^2/z, {z, 1, Infinity}]
    Out[1] = -EulerGamma/2 - Log[2]/2 + (EulerGamma - CosIntegral[2] + Log[2])/2

    In[2] := $Version
    Out[2]=  "4.1 for Microsoft Windows (November 2, 2000)"

which is wrong while Mathematica 4.2 works excellent

    In[1] := Integrate[Cos[z]^2/z, {z, 1, Infinity}]
    Out[1] = Integrate::"idiv": "Integral of"..."does not converge on {1, Infinity)."

    In[2] := $Version
    Out[2]=  "4.2 for Microsoft Windows (February 28, 2002)"


Best wishes,

Vladimir Bondarenko
Mathematical Director
Symbolic Testing Group
Email:  vvb at mail.strace.net

Web  :  http://www.CAS-testing.org/

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