An exact simplification challenge - 100 (EllipticE, GAMMA)
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From: Vladimir Bondarenko on 2 Aug 2010 06:23 Hello, Mathematica: I - I Sqrt[2 Pi] EllipticE[2]/Gamma[3/4]^2 Maple: I-I*sqrt(2*Pi)*EllipticE(sqrt(2))/GAMMA(3/4)^2 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 2 Aug 2010 07:34 Vladimir Bondarenko wrote: > Hello, > > Mathematica: > > I - I Sqrt[2 Pi] EllipticE[2]/Gamma[3/4]^2 > > Maple: > > I-I*sqrt(2*Pi)*EllipticE(sqrt(2))/GAMMA(3/4)^2 > > Can you "elementarize" this ? Numerics suggest it equals 1, thus one wants to show: sqrt(2*Pi)*EllipticE(sqrt(2))= (1+I)*GAMMA(3/4)^2. Writing EllipticE as integral shows, that one has the following identity: sqrt(2)* EllipticE(sqrt(2)) = '(1+I)*(2*EllipticE(1/sqrt(2))-EllipticK(1/sqrt(2)))' Or otherwise said: the following should hold true (2*EllipticE(1/sqrt(2))-EllipticK(1/sqrt(2))) = GAMMA(3/4)^2/sqrt(Pi); Converting to hypergeometrics and simplifying does it: convert(%, hypergeom); simplify(%); is(%); true PS: it does not make *any* sense to post such to the numerical or Matlab group, as you want _non-numerical_ answers.
From: Vladimir Bondarenko on 3 Aug 2010 21:10
On Aug 2, 2:34 pm, Axel Vogt <&nore...(a)axelvogt.de> wrote: > Vladimir Bondarenko wrote: > > Hello, > > > Mathematica: > > > I - I Sqrt[2 Pi] EllipticE[2]/Gamma[3/4]^2 > > > Maple: > > > I-I*sqrt(2*Pi)*EllipticE(sqrt(2))/GAMMA(3/4)^2 > > > Can you "elementarize" this ? > > Numerics suggest it equals 1, thus one wants to show: > > sqrt(2*Pi)*EllipticE(sqrt(2))= (1+I)*GAMMA(3/4)^2. > > Writing EllipticE as integral shows, that one has the > following identity: sqrt(2)* EllipticE(sqrt(2)) = > '(1+I)*(2*EllipticE(1/sqrt(2))-EllipticK(1/sqrt(2)))' > > Or otherwise said: the following should hold true > > (2*EllipticE(1/sqrt(2))-EllipticK(1/sqrt(2))) = > GAMMA(3/4)^2/sqrt(Pi); > > Converting to hypergeometrics and simplifying does it: > > convert(%, hypergeom); > simplify(%); > is(%); > > true > > PS: it does not make *any* sense to post such to the > numerical or Matlab group, as you want _non-numerical_ > answers. "PS: it does not make *any* sense to post such to the numerical or Matlab group, as you want _non-numerical_ answers." Yes, the symbolic ones. We own a MATLAB 7 copy from the MathWorks, there is a symbolic toolboox in MATLAB. Now it is presented by MuPAD 4. Also, we discovered that there are readers in sci.math.num-analysis who by some reasons are interested in symbolic answers... :) |