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Re: Mind+Mathematica


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? 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


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