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Science Forum Index » Math - Symbolic Forum » An exact 1-D integration challenge - 40
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| Author |
Message |
| Vladimir Bondarenko |
 Posted: Sun Jun 03, 2007 1:36 am |
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Guest
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Hello computer algebra buffs,
Surprisingly, none of the computer algebra systems of the
current generation can calculate this integral directly
int(tanh(z)^2/z^2, z= 0..infinity);
Is there a soul who can come up with and display a set of
a CAS commands yielding the exact value of this integral?
Best wishes,
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing |
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| dimitris |
| Posted: Sun Jun 03, 2007 6:04 am |
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Hello Vladimir.
I must admit I have missed all of challenges the last period.
It's good to have you back well and active.
I hope your problems were solved!
Anyway...
As always you asked about a series of steps in CAS.
I use Mathematica and I don't check what Maple do.
But I also you my brain (the little I have!) aditionally!
The trick is to use integration by parts.
(First I examined a little the integrand; no problems in the
integration range)
That is, we have
In[42]:=
HoldForm[Integrate[Tanh[z]^2/z^2, {z, 0, Infinity}] ==
Integrate[Tanh[z]^2*D[-z^(-1), z], {z, 0, Infinity}] ==
Limit[-(Tanh[z]^2/z), z -> Infinity] - Limit[-(Tanh[z]^2/z), z ->
0] + Integrate[D[Tanh[z]^2, z]*(1/z), {z, 0, Infinity}]]
So the original integral is equal to
In[45]:=
Limit[-(Tanh[z]^2/z), z -> Infinity] - Limit[-(Tanh[z]^2/z), z -> 0] +
Integrate[D[Tanh[z]^2, z]/z, {z, 0, Infinity}]
Out[45]=
(14*Zeta[3])/Pi^2
Check now
In[51]:=
N[(14*Zeta[3])/Pi^2,20]
Out[51]=
1.7051135952700231637
In[53]:=
NIntegrate[Tanh[z]^2/z^2, {z, 0, Infinity}, WorkingPrecision -> 30,
PrecisionGoal -> 20]
Out[53]=
1.70511359527002316369
Cheers
Dimitris
/ Vladimir Bondarenko :
Quote: Hello computer algebra buffs,
Surprisingly, none of the computer algebra systems of the
current generation can calculate this integral directly
int(tanh(z)^2/z^2, z= 0..infinity);
Is there a soul who can come up with and display a set of
a CAS commands yielding the exact value of this integral?
Best wishes,
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing |
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