Re: Numerical integration

*To*: mathgroup at smc.vnet.net*Subject*: [mg79675] Re: [mg79621] Numerical integration*From*: Bob Hanlon <hanlonr at cox.net>*Date*: Wed, 1 Aug 2007 05:12:22 -0400 (EDT)*Reply-to*: hanlonr at cox.net

Version 6 appears to have a problem with this integral. $Version 6.0 for Mac OS X x86 (32-bit) (June 19, 2007) expr1 = Integrate[Log[1 + z^2]*(BesselJ[1, z]^2/z), {z, 0, Infinity}] (1/2)*HypergeometricPFQ[{1/2}, {1, 2}, 1]*(1 - 2*EulerGamma + Log[4]) expr2 = expr1 // FunctionExpand // Simplify (-(1/2))*(BesselI[0, 1]^2 - BesselI[1, 1]^2)* (-1 + 2*EulerGamma - Log[4]) expr1 // N 0.790559 expr2 // N 0.790559 NIntegrate[Log[1 + z^2]*(BesselJ[1, z]^2/z), {z, 0, Infinity}] 0.869058 Bob Hanlon ---- dimitris <dimmechan at yahoo.com> wrote: > In[3]:= > Integrate[Log[1 + z^2]*(BesselJ[1, z]^2/z), {z, 0, Infinity}] > N[%, 10] > > Out[3]= > MeijerG[{{1/2}, {}}, {{0, 0}, {-1}}, 1]/(2*Sqrt[Pi]) > > Out[4]= > 0.8732180258611361020606751916`10. > > On another CAS I took, > > convert("Integrate[Log[1 + z^2]*(BesselJ[1, z]^2/z), {z, 0, > Infinity}]",FromMma,evaluate); > evalf(%,20); > > 1/2 > 1/2 (2 Pi BesselI(0, 1) BesselK(0, 1) > > 1/2 / 1/2 > + 2 Pi BesselK(1, 1) BesselI(1, 1)) / Pi > / > > 0.87321802586113613925 > > Both CAS return the same symbolic result. > [An interesting challenge is to simplify the MeijerG > output of Mathematica to that of the other CAS] > > I want to check this symbolic result with NIntegrate. > I have "played around" with the options but I could get > "more closely" than > > In[16]:= > NIntegrate[Log[1+z^2]*(BesselJ[1,z]^2/z),{z, > 0, },MaxRecursion\[Rule]18]//InputForm > > >From In[16]:= > \!\(\* > RowBox[{\(NIntegrate::"slwcon > "\), \(\(:\)\(\ \)\), "\<\"Numerical integration > converging too slowly; suspect > one of the following: singularity, value of the integration being > 0, \ > oscillatory integrand, or insufficient WorkingPrecision. If your > integrand is \ > oscillatory try using the option Method->Oscillatory in NIntegrate. \ > \\!\\(\\*ButtonBox[\\\"More...\\\", ButtonStyle->\\\"RefGuideLinkText\\ > \", \ > ButtonFrame->None, ButtonData:>\\\"NIntegrate::slwcon\\\"]\\)\"\>"}]\) > > >From In[16]:= > \!\(\* > RowBox[{\(NIntegrate::"ncvb"\), \(\(:\)\(\ \)\), "\<\"NIntegrate > failed > to converge to prescribed accuracy > after \\!\\(19\\) recursive bisections in \\!\\(z\\) near \\!\\(z\ > \) = \ > \\!\\(35857.55603944016`\\). \\!\\(\\*ButtonBox[\\\"More...\\\", \ > ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ > ButtonData:>\\\"NIntegrate::ncvb\\\"]\\)\"\>"}]\) > > Out[16]//InputForm= > 0.8732193803103058 > > I would be very happy if someone pointed a tactic for perfroming > satisfactory numerical integration with mathematica for this integral. > I use mathematica 5.2 but you can use Mathematica 6 as well! > > The integral arose in another forum. There it was pointed out that > the performance of Mathematica 6 is buggy as regards numerical > integration. > > I look forward to seeing any replies. > > Greetings from burning Greece! > > Dimitris > > PS > > See here > > http://groups.google.gr/group/sci.math.symbolic/browse_thread/thread/57af36ff6f540a0d/a076ffbc412f974a?hl=el#a076ffbc412f974a > > for above mentioned thread. > >