An exact simplification challenge - 102 (Psi, polylog)
in [Math]
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From: clicliclic on 9 Aug 2010 08:49 Vladimir Bondarenko schrieb: > On Aug 9, 12:42 am, cliclic...(a)freenet.de wrote: > > Vladimir Bondarenko schrieb: > > > > > Mathematica: > > > > > PolyGamma[1, 1/6] + 9 PolyGamma[1, 1/3] - > > > 9 PolyGamma[1, 2/3] - PolyGamma[1, 5/6] - > > > 48 I Sqrt[3] PolyLog[2, ((2 + 3 I) - (1 + 2 I) Sqrt[3])/2] + > > > 48 I Sqrt[3] PolyLog[2, (1 - 3/2 I) - (1/2 - I) Sqrt[3]] > > > > > Maple: > > > > > Psi(1,1/6)+9*Psi(1,1/3)-9*Psi(1,2/3)-Psi(1,5/6)- > > > 48*I*sqrt(3)*polylog(2,1+3/2*I-(1/2+I)*sqrt(3))+ > > > 48*I*sqrt(3)*polylog(2,1-3/2*I+(-1/2+I)*sqrt(3)) > > > > > Can you "elementarize" this ? > > > > The zeta function part > > > > ZETA(2,1/6) + 9*ZETA(2,1/3) - 9*ZETA(2,2/3) - ZETA(2,5/6) > > > > 98.44410402 > > > > may be simplified to > > > > 28*ZETA(2,1/3) - 56/3*pi^2 > > > > 98.44410402 > > > > which corresponds to > > > > -#i*SQRT(3)*(84*LI2((-1+SQRT(3)*#i)/2) + 14*pi^2/3) > > > > 98.44410402 > > > > Oh, I suspect some definitions in Mathematica/Maple > Derive and could be different... 8-( > > In fact, the above expressions in Mathematica/Maple > approximate to > > 57.32873750797116359667145471557897409239645761367... > A private message by Vladimir confirms that nothing is wrong in my interpretation and transformations of his PolyGamma[] or Psi() terms. My LI2() is equivalent to MMA's PolyLog[2,] and Maple's polylog(2,). For the purpose of numerical evaluation and plotting I define this simply as LI2(z) := - INT(LN(1 - t_*z)/t_, t_, 0, 1); in fact, I hardly ever use Derive's library definitions of special functions, I usually substitute my own. (Derive 6.10 has a rudimentary DILOG() in its kernel, which is subject to some simplifying transformations like DILOG(SQRT(5)/2+1/2) -> LN(SQRT(5)/2+1/2)^2/2 - pi^2/15 or DILOG(x) -> -DILOG(1-x) - LN(x)*LN(1-x) + pi^2/6, but numerical evaluation is slow and limited to real numbers.) Martin.
From: clicliclic on 9 Aug 2010 16:34 clicliclic(a)freenet.de schrieb: > > Vladimir Bondarenko schrieb: > > > > Mathematica: > > > > PolyGamma[1, 1/6] + 9 PolyGamma[1, 1/3] - > > 9 PolyGamma[1, 2/3] - PolyGamma[1, 5/6] - > > 48 I Sqrt[3] PolyLog[2, ((2 + 3 I) - (1 + 2 I) Sqrt[3])/2] + > > 48 I Sqrt[3] PolyLog[2, (1 - 3/2 I) - (1/2 - I) Sqrt[3]] > > > > Maple: > > > > Psi(1,1/6)+9*Psi(1,1/3)-9*Psi(1,2/3)-Psi(1,5/6)- > > 48*I*sqrt(3)*polylog(2,1+3/2*I-(1/2+I)*sqrt(3))+ > > 48*I*sqrt(3)*polylog(2,1-3/2*I+(-1/2+I)*sqrt(3)) > > > > Can you "elementarize" this ? > > > > The zeta function part > > ZETA(2,1/6) + 9*ZETA(2,1/3) - 9*ZETA(2,2/3) - ZETA(2,5/6) > > 98.44410402 > > may be simplified to > > 28*ZETA(2,1/3) - 56/3*pi^2 > > 98.44410402 > > which corresponds to > > -#i*SQRT(3)*(84*LI2((-1+SQRT(3)*#i)/2) + 14*pi^2/3) > > 98.44410402 > Having gotten rid of the zeta functions, we are now left with three complex dilogarithm terms: #i*SQRT(3)*(-84*LI2((-1+SQRT(3)*#i)/2)-48*LI2(-SQRT(3)/2+1+#i*(3~ /2-SQRT(3)))+48*LI2(-SQRT(3)/2+1+#i*(SQRT(3)-3/2))-14*pi^2/3) 57.32873750-4.585505421*10^(-10)*#i And these simplify to: 8*SQRT(3)*pi*LN(SQRT(3)+2) [PrecisionDigits:=50,NotationDigits:=50] 57.328737507971163596671454715578974092396457613676 :p Martin.
From: Vladimir Bondarenko on 10 Aug 2010 03:22
On Aug 9, 11:34 pm, cliclic...(a)freenet.de wrote: > cliclic...(a)freenet.de schrieb: > > > > > > > > > Vladimir Bondarenko schrieb: > > > > Mathematica: > > > > PolyGamma[1, 1/6] + 9 PolyGamma[1, 1/3] - > > > 9 PolyGamma[1, 2/3] - PolyGamma[1, 5/6] - > > > 48 I Sqrt[3] PolyLog[2, ((2 + 3 I) - (1 + 2 I) Sqrt[3])/2] + > > > 48 I Sqrt[3] PolyLog[2, (1 - 3/2 I) - (1/2 - I) Sqrt[3]] > > > > Maple: > > > > Psi(1,1/6)+9*Psi(1,1/3)-9*Psi(1,2/3)-Psi(1,5/6)- > > > 48*I*sqrt(3)*polylog(2,1+3/2*I-(1/2+I)*sqrt(3))+ > > > 48*I*sqrt(3)*polylog(2,1-3/2*I+(-1/2+I)*sqrt(3)) > > > > Can you "elementarize" this ? > > > The zeta function part > > > ZETA(2,1/6) + 9*ZETA(2,1/3) - 9*ZETA(2,2/3) - ZETA(2,5/6) > > > 98.44410402 > > > may be simplified to > > > 28*ZETA(2,1/3) - 56/3*pi^2 > > > 98.44410402 > > > which corresponds to > > > -#i*SQRT(3)*(84*LI2((-1+SQRT(3)*#i)/2) + 14*pi^2/3) > > > 98.44410402 > > Having gotten rid of the zeta functions, we are now left with three > complex dilogarithm terms: > > #i*SQRT(3)*(-84*LI2((-1+SQRT(3)*#i)/2)-48*LI2(-SQRT(3)/2+1+#i*(3~ > /2-SQRT(3)))+48*LI2(-SQRT(3)/2+1+#i*(SQRT(3)-3/2))-14*pi^2/3) > > 57.32873750-4.585505421*10^(-10)*#i > > And these simplify to: > > 8*SQRT(3)*pi*LN(SQRT(3)+2) > > [PrecisionDigits:=50,NotationDigits:=50] > > 57.328737507971163596671454715578974092396457613676 > > :p > > Martin. Pefect! So as we see, there are lots of eerie special functions linear combinations which are just "school quantities", very small in size at that as compared with the original expressions. http://groups.google.com/group/sci.math.symbolic/browse_thread/thread/2b7878ea9db46b27/ea861b72145e02aa?#ea861b72145e02aa On the nature and goal of Cyber Tester's challenges [ ... ] "By requesting to publish the explicit human-based solutions we hope to create, to accumulate a solid stock of patterns to be possibly used along with regular algorithms to handle various math tasks automatically." [ ... ] So it would be nice to enjoy your full processing. Cheers, Vladimir |