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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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