Re: Bug in Mathematica 6 - N - 8 (MeijerG, running after 100,000 seconds)

On 11 , 03:22, Vladimir Bondarenko <v...@xxxxxxxxxxxxxxx> wrote:
Our little demo continues... Hello again from the VM
machine which is still ignored by CAS manufactureres.

(* Yet another case of numeric approximation failure *)

(* At a Core 2 duo 2.13 GHz based machine, Mathematica 6 *)
(* keeps running w/o any message after 100,000 seconds *)

N[MeijerG[{{1/2, 1, 1, 1}, {}}, {{1/2, 1/2, 1}, {0}}, I, 1/2]]

For example,

N[Integrate[ArcCsc[z]/(1 + z), {z, 0, Infinity}]]

Best wishes,

Vladimir Bondarenko

VM and GEMM architect
Co-founder, CEO, Mathematical Director

http://www.cybertester.com/ Cyber Tester, LLChttp://maple.bug-list.org/ Maple Bugs Encyclopaediahttp://www.CAS-testing.org/ CAS Testing

It should be mentioned also that Mma 5.2 result is not correct for
the mentioned integral or to be more specific the result is half
right!

In[11]:=
Integrate[ArcCsc[z]/(1 + z), {z, 0, Infinity}]
{N[%], NIntegrate[ArcCsc[z]/(1 + z), {z, 0, Infinity}, MaxRecursion ->
16, MinRecursion -> 12, WorkingPrecision -> 30,
PrecisionGoal -> 6]}

Out[11]=
1 + (2/9)*HypergeometricPFQ[{1, 3/2, 2}, {5/2, 5/2}, 1]
Out[12]=
{1.831931188354452, 1.8319311875276281536`6.326601825621108 -
1.2337003425957649583`6.154902354638181*I}

Moreover direct application of the NL formula (the returned
antiderivative
is a smooth function in the integration range) gives rise to another
bug
(in version 5.2)

In[22]:=
F = Integrate[ArcCsc[z]/(1 + z), z]
Plot[Re[F], {z, 0, 10}]
Plot[Im[F], {z, 0, 10}, PlotRange -> {-2, 0}]
Limit[F, z -> Infinity] - Limit[F, z -> 0, Direction -> -1]

Out[22]=
(1/2)*I*Pi*ArcCsc[z] + 2*Pi*Log[1 + E^((-I)*ArcCsc[z])] + Pi*Log[1 -
I*E^(I*ArcCsc[z])] +
2*ArcCsc[z]*Log[1 - I*E^(I*ArcCsc[z])] - ArcCsc[z]*Log[1 -
E^(2*I*ArcCsc[z])] - 2*Pi*Log[Cos[ArcCsc[z]/2]] -
Pi*Log[Sin[(1/4)*(Pi + 2*ArcCsc[z])]] - 2*I*PolyLog[2,
I*E^(I*ArcCsc[z])] + (1/2)*I*PolyLog[2, E^(2*I*ArcCsc[z])]

Out[23]=
Graphics

Out[24]=
Graphics

Out[25]=
Infinity

Watch the infinity in the result.

Mma (5.2) evaluates incorrectly the limit at 0 unfortunately.

In[29]:=
FullSimplify[(Limit[F, z -> #1] & ) /@ {0, Infinity}]
Out[29]=
{-Infinity, 2*Catalan - (I*Pi^2)/8 + Pi*Log[8]}


Dimitris




.