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Science Forum Index » Math - Symbolic Forum » An exact simplification challenge - 60 (LerchPhi,...
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| Vladimir Bondarenko... |
 Posted: Sat May 10, 2008 6:09 am |
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Hello CAS Fan the Earthling,
Caramba! The Final Battle might break soon! Are WE the Earthlings
ready?
Forget the morons, don't basti your diamond time to the drivel;
instead, get equipped with extra flexibility to fight for our
absolute valuables, namely, liberty, beverages and eatables!
Train h a r d, fight e a s y! - The genius buzzwords fearless
and resourceful and victorious Field Marshal Suvorov reiterated
each morning to his soldiers, and won battle after battle!
The eerie (striped!) Computers are coming! Train yourself, hone
your bean, don't let your quick little gray cells to get fat and
sluggish...
So, is there an Audacious Warrior the Simplifier to come up with
a sequence of CAS commands to "elementarize" this
LerchPhi(1/9, 2, 1/2) - 3*dilog(3/4)
?
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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| Posted: Sat May 10, 2008 12:02 pm |
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Vladimir Bondarenko schrieb:
Quote:
Hello CAS Fan the Earthling,
[...] is there an Audacious Warrior the Simplifier to come up with
a sequence of CAS commands to "elementarize" this
LerchPhi(1/9, 2, 1/2) - 3*dilog(3/4)
Derive 8.07 handles this automatically in a fraction of a second as
follows:
LERCH_PHI(1/9,2,1/2)-3*DILOG(3/4)
" -> "
2*(LERCH_PHI(1/3,2,1)+LERCH_PHI(-1/3,2,1))-3*DILOG(3/4)
" -> "
2*(DILOG(2/3)/(1/3)+DILOG(4/3)/(-1/3))-3*DILOG(3/4)
" -> "
2*(3*(-DILOG(3/2)-LN(2/3)^2/2)-3*DILOG(4/3))-3*DILOG(3/4)
" -> "
-6*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2+6*LN(2)*LN(3)-3*LN(2)^2-3*(-
DILOG(4/3)-L~
N(3/4)^2/2)
" -> "
-6*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2+6*LN(2)*LN(3)-3*LN(2)^2-3*(-
DILOG(4/3)-(~
LN(3)-2*LN(2))^2/2)
-3*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2/2+3*LN(2)^2
Numerically, this approximates to 3.248520221. There are obvious
problems with the rule strings here. Also note that Derive assumes
dilog(z) = Li_2(1-z); cf. Wikipedia at <http://en.wikipedia.org/wiki/
Polylogarithm>.
Martin. |
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| Vladimir Bondarenko... |
| Posted: Sat May 10, 2008 4:52 pm |
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David W. Cantrell <DWCantr... at (no spam) sigmaxi.net> writes:
DWC> Fascinating! Where can I get a copy of Derive 8.07 ?
Same question here.
I would buy Derive 8.07 for $2000 without a fraction of
second of hesitation.
On May 10, 3:35 pm, David W. Cantrell <DWCantr... at (no spam) sigmaxi.net> wrote:
Quote: cliclic... at (no spam) freenet.de wrote:
Vladimir Bondarenko schrieb:
Hello CAS Fan the Earthling,
[...] is there an Audacious Warrior the Simplifier to come up with
a sequence of CAS commands to "elementarize" this
LerchPhi(1/9, 2, 1/2) - 3*dilog(3/4)
Derive 8.07
Fascinating! Where can I get a copy of Derive 8.07 ?
David
handles this automatically in a fraction of a second as follows:
LERCH_PHI(1/9,2,1/2)-3*DILOG(3/4)
" -> "
2*(LERCH_PHI(1/3,2,1)+LERCH_PHI(-1/3,2,1))-3*DILOG(3/4)
" -> "
2*(DILOG(2/3)/(1/3)+DILOG(4/3)/(-1/3))-3*DILOG(3/4)
" -> "
2*(3*(-DILOG(3/2)-LN(2/3)^2/2)-3*DILOG(4/3))-3*DILOG(3/4)
" -> "
-6*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2+6*LN(2)*LN(3)-3*LN(2)^2-3*(-
DILOG(4/3)-L~
N(3/4)^2/2)
" -> "
-6*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2+6*LN(2)*LN(3)-3*LN(2)^2-3*(-
DILOG(4/3)-(~
LN(3)-2*LN(2))^2/2)
-3*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2/2+3*LN(2)^2
Numerically, this approximates to 3.248520221. There are obvious
problems with the rule strings here. Also note that Derive assumes
dilog(z) = Li_2(1-z); cf. Wikipedia at <http://en.wikipedia.org/wiki/
Polylogarithm>.
Martin.- Hide quoted text -
- Show quoted text - |
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| Vladimir Bondarenko... |
| Posted: Sat May 10, 2008 4:58 pm |
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C> -3*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2/2+3*LN(2)^2
I believe dilog is not elementary :)
Could you simplify this further to, say, logarithms?
On May 10, 3:02 pm, cliclic... at (no spam) freenet.de wrote:
Quote: Vladimir Bondarenko schrieb:
Hello CAS Fan the Earthling,
[...] is there an Audacious Warrior the Simplifier to come up with
a sequence of CAS commands to "elementarize" this
LerchPhi(1/9, 2, 1/2) - 3*dilog(3/4)
Derive 8.07 handles this automatically in a fraction of a second as
follows:
LERCH_PHI(1/9,2,1/2)-3*DILOG(3/4)
" -> "
2*(LERCH_PHI(1/3,2,1)+LERCH_PHI(-1/3,2,1))-3*DILOG(3/4)
" -> "
2*(DILOG(2/3)/(1/3)+DILOG(4/3)/(-1/3))-3*DILOG(3/4)
" -> "
2*(3*(-DILOG(3/2)-LN(2/3)^2/2)-3*DILOG(4/3))-3*DILOG(3/4)
" -> "
-6*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2+6*LN(2)*LN(3)-3*LN(2)^2-3*(-
DILOG(4/3)-L~
N(3/4)^2/2)
" -> "
-6*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2+6*LN(2)*LN(3)-3*LN(2)^2-3*(-
DILOG(4/3)-(~
LN(3)-2*LN(2))^2/2)
-3*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2/2+3*LN(2)^2
Numerically, this approximates to 3.248520221. There are obvious
problems with the rule strings here. Also note that Derive assumes
dilog(z) = Li_2(1-z); cf. Wikipedia at <http://en.wikipedia.org/wiki/
Polylogarithm>.
Martin. |
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| David W. Cantrell... |
| Posted: Sat May 10, 2008 5:35 pm |
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Guest
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clicliclic at (no spam) freenet.de wrote:
Quote: Vladimir Bondarenko schrieb:
Hello CAS Fan the Earthling,
[...] is there an Audacious Warrior the Simplifier to come up with
a sequence of CAS commands to "elementarize" this
LerchPhi(1/9, 2, 1/2) - 3*dilog(3/4)
Derive 8.07
Fascinating! Where can I get a copy of Derive 8.07 ?
David
Quote: handles this automatically in a fraction of a second as follows:
LERCH_PHI(1/9,2,1/2)-3*DILOG(3/4)
" -> "
2*(LERCH_PHI(1/3,2,1)+LERCH_PHI(-1/3,2,1))-3*DILOG(3/4)
" -> "
2*(DILOG(2/3)/(1/3)+DILOG(4/3)/(-1/3))-3*DILOG(3/4)
" -> "
2*(3*(-DILOG(3/2)-LN(2/3)^2/2)-3*DILOG(4/3))-3*DILOG(3/4)
" -> "
-6*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2+6*LN(2)*LN(3)-3*LN(2)^2-3*(-
DILOG(4/3)-L~
N(3/4)^2/2)
" -> "
-6*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2+6*LN(2)*LN(3)-3*LN(2)^2-3*(-
DILOG(4/3)-(~
LN(3)-2*LN(2))^2/2)
-3*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2/2+3*LN(2)^2
Numerically, this approximates to 3.248520221. There are obvious
problems with the rule strings here. Also note that Derive assumes
dilog(z) = Li_2(1-z); cf. Wikipedia at <http://en.wikipedia.org/wiki/
Polylogarithm>.
Martin. |
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| sashap... |
| Posted: Sat May 10, 2008 6:39 pm |
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On May 10, 9:58 pm, Vladimir Bondarenko <v... at (no spam) cybertester.com> wrote:
Quote: C> -3*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2/2+3*LN(2)^2
I believe dilog is not elementary 
In[33]:= ((2 LerchPhi[1/3, 2, 1] + 2 LerchPhi[-(1/3), 2, 1]) -
3 PolyLog[2, 1/4] // FullSimplify) /. {PolyLog[2, 1/4] :>
2 PolyLog[2, 1/2] + 2 PolyLog[2, -(1/2)]} //
Expand // FullSimplify
Out[33]= 1/2 (\[Pi]^2 - 24 ArcCoth[5] Log[2])
In[34]:= N[%, 50]
Out[34]= 3.2485202215410339388451243568790420841863576730735
Quote:
Could you simplify this further to, say, logarithms?
On May 10, 3:02 pm, cliclic... at (no spam) freenet.de wrote:
Vladimir Bondarenko schrieb:
Hello CAS Fan the Earthling,
[...] is there an Audacious Warrior the Simplifier to come up with
a sequence of CAS commands to "elementarize" this
LerchPhi(1/9, 2, 1/2) - 3*dilog(3/4)
Derive 8.07 handles this automatically in a fraction of a second as
follows:
LERCH_PHI(1/9,2,1/2)-3*DILOG(3/4)
" -> "
2*(LERCH_PHI(1/3,2,1)+LERCH_PHI(-1/3,2,1))-3*DILOG(3/4)
" -> "
2*(DILOG(2/3)/(1/3)+DILOG(4/3)/(-1/3))-3*DILOG(3/4)
" -> "
2*(3*(-DILOG(3/2)-LN(2/3)^2/2)-3*DILOG(4/3))-3*DILOG(3/4)
" -> "
-6*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2+6*LN(2)*LN(3)-3*LN(2)^2-3*(-
DILOG(4/3)-L~
N(3/4)^2/2)
" -> "
-6*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2+6*LN(2)*LN(3)-3*LN(2)^2-3*(-
DILOG(4/3)-(~
LN(3)-2*LN(2))^2/2)
-3*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2/2+3*LN(2)^2
Numerically, this approximates to 3.248520221. There are obvious
problems with the rule strings here. Also note that Derive assumes
dilog(z) = Li_2(1-z); cf. Wikipedia at <http://en.wikipedia.org/wiki/
Polylogarithm>.
Martin. |
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| Posted: Sun May 11, 2008 9:10 pm |
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Vladimir Bondarenko schrieb:
Quote:
-3*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2/2+3*LN(2)^2
I believe dilog is not elementary :)
Could you simplify this further to, say, logarithms?
Here's a suggestion for another challenge (you may take it as a
counterchallenge by striped invaders as well):
Can this expression (involving the Legendre elliptic integrals
F(phi,k), E(phi,k), K(k), E(k)) be simplified to a significant degree?
(E(arctan(t), k) + E(arccot(sqrt(1-k^2)*t), k)) * K(k) -
(F(arctan(t), k) + F(arccot(sqrt(1-k^2)*t), k)) * E(k)
Hope this doesn't turn out too easy.
Martin. |
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| Vladimir Bondarenko... |
| Posted: Sun May 11, 2008 9:38 pm |
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I was wondering what the definition for
F(phi,k), E(phi,k), K(k), E(k))
do you use?
Maple's one? Mathematica's one? Anything else?
On May 12, 12:10 am, cliclic... at (no spam) freenet.de wrote:
Quote: Vladimir Bondarenko schrieb:
-3*DILOG(4/3)-6*DILOG(3/2)-3*LN(3)^2/2+3*LN(2)^2
I believe dilog is not elementary :)
Could you simplify this further to, say, logarithms?
Here's a suggestion for another challenge (you may take it as a
counterchallenge by striped invaders as well):
Can this expression (involving the Legendre elliptic integrals
F(phi,k), E(phi,k), K(k), E(k)) be simplified to a significant degree?
(E(arctan(t), k) + E(arccot(sqrt(1-k^2)*t), k)) * K(k) -
(F(arctan(t), k) + F(arccot(sqrt(1-k^2)*t), k)) * E(k)
Hope this doesn't turn out too easy.
Martin. |
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| Posted: Sun May 11, 2008 10:39 pm |
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Vladimir Bondarenko schrieb:
Quote: I was wondering what the definition for
F(phi,k), E(phi,k), K(k), E(k))
do you use?
Maple's one? Mathematica's one? Anything else?
Sorry! The Gradshteyn-Ryzhik one, e.g.
F(phi, k) = int(1/sqrt(1-k^2·SIN(t)^2), t, 0, phi),
etc. This is a vital piece of information, of course!
Martin. |
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