Re: Mind+Mathematica

*To*: mathgroup at smc.vnet.net*Subject*: [mg78262] Re: Mind+Mathematica*From*: "Jean-Marc Gulliet" <jeanmarc.gulliet at gmail.com>*Date*: Wed, 27 Jun 2007 05:25:06 -0400 (EDT)*References*: <200706210945.FAA26122@smc.vnet.net> <f5gat7$gc8$1@smc.vnet.net>

David W.Cantrell wrote: > 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? Yes. Here is the evaluation on my system: In[1]:= Integrate[Sin[z]*Sin[z^3 + z], {z, 0, Infinity}] During evaluation of In[1]:= Integrate::idiv: Integral of Sin[z] \ Sin[z+z^3] does not converge on {0,\[Infinity]}. >> Out[1]= Integrate[Sin[z]*Sin[z + z^3], {z, 0, Infinity}] In[2]:= $Version Out[2]= "6.0 for Microsoft Windows (32-bit) (April 20, 2007)" > 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 Regards, Jean-Marc

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