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!

**Follow-Ups**:**Re: Mind+Mathematica***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>