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Re: Mind+Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg78116] Re: Mind+Mathematica
• From: c <flutzpah at gmail.com>
• Date: Sat, 23 Jun 2007 07:09:44 -0400 (EDT)
• References: <200706210945.FAA26122@smc.vnet.net><f5gat7$gc8$1@smc.vnet.net>

On Jun 22, 5:13 am, Andrzej Kozlowski <a... 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 ?
>
> Andrzej Kozlowski

Again, somehow our versions seem to be giving different _symbolic_
answers:

In[1]:= $Version

Out[1]= "6.0 for Linux x86 (32-bit) (April 20, 2007)"

In[2]:= Integrate[TrigToExp[Sin[z]*Sin[z^3 + z]], {z, 0, Infinity}]
N[%]

Out[2]= (-(1/4))*Pi*((2*AiryAi[2/3^(1/3)])/3^(1/3) +
      2/Gamma[-(1/3)])

Out[3]= 0.2957412258497819

       Curtis O.  (cfo at lanl.gov, normally)

• References:
  • Mind+Mathematica
    • From: dimitris <dimmechan@yahoo.com>