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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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