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

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

but Mathematica 4.2 handles correctly

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

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

Even simpler,

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

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

which is wrong while Mathematica 4.2 works excellent

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

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

Best wishes,

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

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

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```

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