An exact simplification challenge - 102 (Psi, polylog)
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From: Vladimir Bondarenko on 4 Aug 2010 21:24 Hello, 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 ? Cheers, Vladimir Bondarenko Co-founder, CEO, Mathematical Director http://www.cybertester.com/ Cyber Tester Ltd. ---------------------------------------------------------------- "We must understand that technologies like these are the way of the future." ---------------------------------------------------------------- ---------------------------------------------------------------- http://groups.google.com/group/sci.math/msg/9f429c3ea5649df5 "...... the challenges imply that a solution is built within the framework of the existent CAS functions & built-in definitions." ---------------------------------------------------------------- ----------------------------------------------------------------
From: Axel Vogt on 5 Aug 2010 15:28 Vladimir Bondarenko wrote: > Hello, > > 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 ? Maple 12 runs into bugs here. a := -2304*ln(xi)*(xi^4+35*3^(1/2)*xi^3-61*xi^3-9*3^(1/2)*xi^2+ 18*xi^2-3^(1/2)*xi-7*xi-4*3^(1/2)+7) b := 1/3840*(15+15*I+(9+7*I)*3^(1/2))/ (-1+2*xi-I*3^(1/2)) Then for Int(a*b, xi=0..1) the symbolic and numerical solutions are quite different, even with increased precision.
From: clicliclic on 8 Aug 2010 17:42 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 Martin.
From: Vladimir Bondarenko on 8 Aug 2010 17:52 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 > > Martin. 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... Cheers, Vladimir
From: clicliclic on 8 Aug 2010 19:09
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 > > > > Martin. > > 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... > > Cheers, Vladimir Please explain the meaning of this part of your expression in terms of Derive's DIGAMMA() or ZETA(). Am I wrong in identifying your PolyGamma[n,q] = Psi(n,q) for n = 1, 2, 3, ... with (-1)^(n+1)*n!*ZETA(n+1,q) ? For DIGAMMA(1/6) + 9*DIGAMMA(1/3) - 9*DIGAMMA(2/3) - DIGAMMA(5/6), I obtain -21.76559237 instead (but these are elementary anyway). I will try to repair this tomorrow once I know what's what. Martin. |