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MathGroup Archive 2007

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Re: Mind+Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg78262] Re: Mind+Mathematica
  • From: "Jean-Marc Gulliet" <jeanmarc.gulliet at gmail.com>
  • Date: Wed, 27 Jun 2007 05:25:06 -0400 (EDT)
  • References: <200706210945.FAA26122@smc.vnet.net> <f5gat7$gc8$1@smc.vnet.net>

David W.Cantrell wrote:
> Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
>> On 21 Jun 2007, at 18:45, dimitris wrote:
>>
>>> The integral
>>>
>>> Integrate[Sin[z]*Sin[z^3 + z], {z, 0, Infinity}]
>>>
>>> (as I was informed)
>>>
>>> gives a incorrectly divergent message.
>>> The integral however is convergent.
>>>
>>> The following is part of my response to another forum.
>>> Demonstrate how vital is to help Mathematica sometimes.
>>>
>>> In[2]:=
>>> $Version
>>>
>>> Out[2]=
>>> "5.2 for Microsoft Windows (June 20, 2005)"
>>>
>>> In[3]:=
>>> int=Integrate[Sin[z]*Sin[z^3 + z], {z, 0, Infinity}](*the integral
>>> stays unevaluated*)
>>>
>>> Out[3]=
>>> Integrate[Sin[z]*Sin[z + z^3], {z, 0, Infinity}]
>>>
>>> In[3]:=
>>> int2 = (int /. Integrate[f_, x_] :> Integrate[#1, {z, 0, Infinity}]
>>> & ) /@ Expand[Sin[z]*TrigExpand[Sin[z^3 + z]]]
>>>
>>> Out[3]=
>>> (1/72)*(2*Sqrt[6]*Pi*(BesselI[1/3, (4*Sqrt[2/3])/3] - BesselJ[1/3,
>>> (4*Sqrt[2/3])/3]) +
>>>     3*Gamma[1/3]*(2*Sqrt[3] - Sqrt[2]*BesselI[-(1/3), (4*Sqrt[2/3])/
>>> 3]*Gamma[2/3] - Sqrt[2]*BesselJ[-(1/3), (4*Sqrt[2/3])/3]*Gamma[2/3]))
>>> + Integrate[Cos[z]*Sin[z]*Sin[z^3], {z, 0, Infinity}]
>>>
>>> In[4]:=
>>> int3 = (1/2)*Integrate[Sin[2*z]*Sin[z^3], {z, 0, Infinity}]
>>>
>>> Out[4]=
>>> (Pi*(AiryAi[-(2/3^(1/3))] - AiryAi[2/3^(1/3)]))/(4*3^(1/3))
>>>
>>> In[5]:=
>>> FullSimplify[int2 /. Integrate[x___] :> int3]
>>>
>>> Out[5]=
>>> (-2*3^(1/6)*Pi*AiryAi[2/3^(1/3)] + Gamma[1/3])/(4*Sqrt[3])
>>>
>>> In[6]:=
>>> N[%, 40]
>>>
>>> Out[6]=
>>> 0.295741225849781931593673891336119670357883693300484102195`40.
>>>
>>> Brought to you by M^2
>>> (Man+Mathematica!)
>>>
>>> Dimitris
>>>
>>> PS
>>> I spent almost two hours to figure out a workaround.
>>> How ancient Greeks said:
>>> "It is not easy to get Goods"
>>>
>>> PS2
>>> Enjoy Mathematics and Mathematica!
>>>
>>>
>> However...
>>
>> Integrate[TrigToExp[Sin[z]*Sin[z^3 + z]], {z, 0, Infinity}]
>>
>> (1/6)*((-Sqrt[2])*BesselK[-(1/3), (4*Sqrt[2/3])/3] - (3*Pi)/Gamma[-
>> (1/3)])
>>
>> N[%, 10]
>> 0.29574122584978190891740677731`10.
>>
>> So who needs Mind when you have Mathematica 6.0 ?
>
> Then I'm curious. What is the result of using just
>
> Integrate[Sin[z]*Sin[z^3 + z], {z, 0, Infinity}]
>
> in version 6? Does it return the integral unevaluated, together with a
> statement about divergence?

Yes. Here is the evaluation on my system:

In[1]:= Integrate[Sin[z]*Sin[z^3 + z], {z, 0, Infinity}]

During evaluation of In[1]:= Integrate::idiv: Integral of Sin[z] \
Sin[z+z^3] does not converge on {0,\[Infinity]}. >>

Out[1]= Integrate[Sin[z]*Sin[z + z^3], {z, 0, Infinity}]

In[2]:= $Version

Out[2]= "6.0 for Microsoft Windows (32-bit) (April 20, 2007)"

> If so, that's a bug. And one would then need to
> use a bit of mind to think that Mathematica 6 might be wrong and that
> TrigToExp might help it get a correct answer.
>
> David

Regards,
Jean-Marc


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