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MathGroup Archive 2007

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Mind+Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg78010] Mind+Mathematica
  • From: dimitris <dimmechan at yahoo.com>
  • Date: Thu, 21 Jun 2007 05:45:59 -0400 (EDT)

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!



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