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/ http://maple.bug-list.org/VER2/ (under tuning) http://maple.bug-list.org/VER3/ (under tuning) http://maple.bug-list.org/VER1/ (under tuning) http://www.beautyriot.com/ (teamwork) http://www.ohaha.com/ (teamwork) Voice: (380)-652-447325 Mon-Fri 9 a.m. - 6 p.m. Mail : 76 Zalesskaya Str, Simferopol, Crimea, Ukraine