Re: Re: Integrate modified in version 6?
- To: mathgroup at smc.vnet.net
- Subject: [mg78096] Re: [mg77976] Re: [mg77938] Integrate modified in version 6?
- From: Bob Hanlon <hanlonr at cox.net>
- Date: Fri, 22 Jun 2007 06:53:41 -0400 (EDT)
- Reply-to: hanlonr at cox.net
I get -Infinity on my machine with v6. I never had a copy of the beta version. $Version 6.0 for Mac OS X x86 (32-bit) (April 20, 2007) Integrate[z*(ArcSin[z]/(1 + z)^2), {z, 0, 1}] -Infinity Bob Hanlon ---- Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote: > My version of Mathematica 6.0 gives: > > Integrate[z*(ArcSin[z]/(1 + z)^2), {z, 0, 1}] > > (1/2)*(MeijerG[{{-(1/2), 0}, {1}}, {{0, 0, 1/2}, {}}, 1] - > MeijerG[{{-(1/2), 0}, {1, 1}}, {{0, 0, 1/2, 1/2}, {}}, 1]/Sqrt > [Pi]) > > I don't know whether this answer is right or wrong (Mathematica takes > for ever to evaluate this numerically) but one thing is clear: it > certainly is not -Infinity. Which only proves, what you would have > already known if you read some past posts on similar topics, namely, > that you can't trust the person you got this information from because > he has a habit of using beta versions long past the release date and > then calling bugs he finds in the beta version as "bugs in > Mathematica". He has done this before (on this forum) and it seems he > is still doing it on others. > > Andrzej Kozlowski > > > On 20 Jun 2007, at 18:30, dimitris wrote: > > > I don't have 6 to be more rigorous but > > based on some integrals post in another > > forum by Vladimir Bondarenko I am quite > > sure that something has change in the integration > > algorithm for definite integrals... > > > > Say for example the integral Integrate[z ArcSin[z]/(1+z)^2, {z, 0, > > 1}]. > > > > In Mathematica 6 we have > > > > Integrate[z ArcSin[z]/(1+z)^2, {z, 0, 1}] > > -Infinity > > > > which is far away from truth. > > > > The good old (?) Mathematica 5.2 returns > > > > In[3]:= > > Integrate[z*(ArcSin[z]/(1 + z)^2), {z, 0, 1}] > > {N[%], NIntegrate[z*(ArcSin[z]/(1 + z)^2), {z, 0, 1}]} > > > > Out[3]= > > -1 - 2*Catalan + Pi*(1/4 + Log[2]) > > Out[4]= > > {0.13105306534661265, 0.1310530653479215} > > > > Let's add an rule for Limit in 5.2 > > > > Unprotect[Limit]; > > Limit[x___] := Null /; Print[InputForm[limit[x]]]; > > > > Let's get the integral > > > > In[4]:= > > Integrate[z*(ArcSin[z]/(1 + z)^2), {z, 0, 1}] > > > >> From In[4]:= > > limit[z^2, z -> 0, Direction -> -1, Assumptions -> True] > >> From In[4]:= > > limit[(1/4 - (-1 + z)^2/16)*(Pi/2 + I*Sqrt[2]*Sqrt[-1 + > > z] - ((I/6)*(-1 + z)^(3/2))/Sqrt[2] + (((3*I)/80)*(-1 + > > z)^(5/2))/ > > Sqrt[2]), z -> 1, Direction -> 1, Assumptions -> True] > >> From In[4]:= > > limit[(1/4 - (-1 + z)^2/16)*(Pi/2 + I*Sqrt[2]*Sqrt[-1 + > > z] - ((I/6)*(-1 + z)^(3/2))/Sqrt[2] + (((3*I)/80)*(-1 + > > z)^(5/2))/ > > Sqrt[2]), z -> 1, Direction -> 1, Assumptions -> True] > >> From In[4]:= > > limit[z^2, z -> 0, Direction -> -1, Assumptions -> True] > >> From In[4]:= > > limit[((-I)*(1 + z)*ArcSin[z]^2 + ArcSin[z]*(2 + I*Pi*(1 + z) + 4*( > > 1 + z)*Log[1 - I*E^( > > I*ArcSin[z])]) + 2*(Sqrt[1 - z^2] + Pi*( > > 1 + z)*Log[1 - I*E^( > > I*ArcSin[z])] + 2*Pi*(1 + z)*Log[(1 + > > E^(I*ArcSin[z]))/ > > E^(I* > > ArcSin[z])] - 2*Pi*Log[Cos[ArcSin[z]/2]] - 2* > > Pi*z*Log[Cos[ArcSin[z]/ > > 2]] - Pi*Log[Sin[(Pi + 2*ArcSin[z])/4]] - Pi*z*Log[ > > Sin[(Pi + > > 2*ArcSin[z])/4]]) - (4* > > I)*(1 + z)*PolyLog[2, I*E^(I*ArcSin[z])])/(2*(1 + > > z)), z -> 1, Direction -> 1, Assumptions -> > > True] > >> From In[4]:= > > limit[((-I)*(1 + z)*ArcSin[z]^2 + ArcSin[z]*(2 + I*Pi*(1 + z) + 4*( > > 1 + z)*Log[1 - I*E^( > > I*ArcSin[z])]) + 2*(Sqrt[1 - z^2] + Pi*( > > 1 + z)*Log[1 - I*E^( > > I*ArcSin[z])] + 2*Pi*(1 + z)*Log[(1 + > > E^(I*ArcSin[z]))/ > > E^(I* > > ArcSin[z])] - 2*Pi*Log[Cos[ArcSin[z]/2]] - 2* > > Pi*z*Log[Cos[ArcSin[z]/ > > 2]] - Pi*Log[Sin[(Pi + 2*ArcSin[z])/4]] - Pi*z*Log[ > > Sin[(Pi + > > 2*ArcSin[z])/4]]) - (4* > > I)*(1 + z)*PolyLog[2, I*E^(I*ArcSin[z])])/(2*(1 + > > z)), z -> 0, Direction -> -1, Assumptions -> > > True] > > > > It can be seen that the integral is evaluated by application of > > the NL formula. > > > > Let's do the same in version 6: > > > > Here is the output as Vladimir sent me > > (sortening a little!) > > > > limit[z^2, z -> 0, Direction -> -1, Assumptions -> True] > > limit[Pi/8 - ((I/2)*Sqrt[-1 + z])/Sqrt[2] + ((I/24)*(-1 + \ > > z)^(3/2))/Sqrt[2] - (Pi*(-1 + z)^2)/32, z -> 1, Direction -> 1, \ > > Assumptions -> True] > > limit[Pi/8 - ((I/2)*Sqrt[-1 + z])/Sqrt[2] + ((I/24)*(-1 + \ > > z)^(3/2))/Sqrt[2] - (Pi*(-1 + z)^2)/32, z -> 1, Direction -> 1, \ > > Assumptions -> True] > > limit[z^2, z -> 0, Direction -> -1, Assumptions -> True] > > > > limit[z*((Pi*Integrate`ImproperDump`MeijerGfunction[{}, {1}, {0}, {}, > > \ > > z^2])/2 - (Sqrt[Pi]*Integrate`ImproperDump`MeijerGfunction[{}, {1, \ > > 1}, {0, 1/2}, {}, \ > > z^2])/2)*Integrate`ImproperDump`MeijerGfunction[{-1}, {}, {0}, {}, \ > > z], z -> Infinity, Assumptions -> True] > > > > limit[2/(z^2*(Pi*Integrate`ImproperDump`MeijerGfunction[{}, {1}, {0}, > > \ > > {}, z^2] - Sqrt[Pi]*Integrate`ImproperDump`MeijerGfunction[{}, {1, \ > > 1}, {0, 1/2}, {}, z^2])*Integrate`ImproperDump`MeijerGfunction[{-1}, > > \ > > {}, {0}, {}, z]), z -> Infinity, Assumptions -> True] > > > > limit[System`MeijerGDump`zz$2982, System`MeijerGDump`zz$2982 -> 1] > > > > limit[(-3*Pi*System`MeijerGDump`zz$2982)/2, \ > > System`MeijerGDump`zz$2982 -> 1] > > > > limit[(Pi*(-Sqrt[System`MeijerGDump`zz$2982] - \ > > ArcTan[Sqrt[System`MeijerGDump`zz$2982]] + \ > > System`MeijerGDump`zz$2982*ArcTan[Sqrt[System`MeijerGDump`zz > > $2982]]))/ > > \ > > (-1 + System`MeijerGDump`zz$2982) - \ > > (3*Pi*System`MeijerGDump`zz$2982*((2*System`MeijerGDump`zz$2982*((1 - > > \ > > System`MeijerGDump`zz$2982)^(-1) - (-System`MeijerGDump`zz$2982 - \ > > Log[1 - System`MeijerGDump`zz$2982])/System`MeijerGDump`zz$2982^2))/3 > > \ > > - Log[1 - System`MeijerGDump`zz$2982]/System`MeijerGDump`zz$2982))/2 > > \ > > + (Pi*(-Log[System`MeijerGDump`zz$2982] + PolyGamma[0, 1/2] - \ > > PolyGamma[0, 3/2]))/2, System`MeijerGDump`zz$2982 -> 1, Direction -> > > \ > > 1] > > > > limit[(2*K$3121*(1 + K$3121)*System`MeijerGDump`zz$3095^(1/2 + \ > > K$3121)*Gamma[1/2 + K$3121])/((1 + 2*K$3121)*Gamma[1 + K$3121]), \ > > K$3121 -> Infinity, Assumptions -> True] > > > > limit[System`MeijerGDump`zz$3095, K$3121 -> Infinity, Analytic -> \ > > True, Assumptions -> K$3121 > 1073741824] > > > > limit[K$3121^(-1), K$3121 -> 0, Assumptions -> K$3121^(-1) > \ > > 1073741824] > > > > limit[K$3121^(-1), K$3121 -> 0, Assumptions -> K$3121^(-1) > \ > > 1073741824] > > > > limit[(2 + K$3121)/(2*K$3121), K$3121 -> 0, Assumptions -> \ > > K$3121^(-1) > 1073741824] > > > > limit[(2 + K$3121)/(2*K$3121), K$3121 -> 0, Assumptions -> \ > > K$3121^(-1) > 1073741824] > > > > limit[((2 + K$3121)*(1 + \ > > 2*K$3121)^2*System`MeijerGDump`zz$3095)/(2*(1 + K$3121)^2*(3 + \ > > 2*K$3121)), K$3121 -> Infinity, Assumptions -> True] > > > > limit[System`MeijerGDump`zz$3095^(1/2 + K$3806)/Sqrt[K$3806], K$3806 - > >> \ > > Infinity, Assumptions -> True] > > > > -Infinity > > > > Bang! > > > >> From this output I am quite sure that no NL thoerem > > to the indefinite integral takes place but rather > > straightly convolution (I write it correct now; > > Cheers David Cantrell!) > > > > Any comments by WRI well informative persons > > (and other of course!) will be greatly appreciate. > > > > > >