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Re: Hypergeometric2F1

  • To: mathgroup at smc.vnet.net
  • Subject: [mg93157] Re: Hypergeometric2F1
  • From: Roland Farnzius <roland.franzius at uos.de>
  • Date: Wed, 29 Oct 2008 05:49:24 -0500 (EST)
  • References: <ge6nfi$li8$1@smc.vnet.net>

Artur wrote:

> Who know which Mathematica procedure to use to find such a,b,c that
> ArcCosh[2]/ArcCosh[2-x]==Hypergeometric2F1[a,b,c,x] for {x,-Infinity,1}

If you want a solution to hold exactly for all x it does not seem to work.

coeff = (List @@
    Series[ArcCosh[2]/ArcCosh[2 - x] -
       Hypergeometric2F1[a, b, c, x], {x, 0, 3}][[3]])

gives the coefficient list of the difference from the Tylor series in x 
at 0:

{-((a*b)/c) + 1/(Sqrt[3]*ArcCosh[2]),
   -((a*(1 + a)*b*(1 + b))/(2*c*(1 + c))) + 1/(3*ArcCosh[2]^2) +
    1/(3*Sqrt[3]*ArcCosh[2]),
   -((a*(1 + a)*(2 + a)*b*(1 + b)*(2 + b))/(6*c*(1 + c)*(2 + c))) +
    1/(3*Sqrt[3]*ArcCosh[2]^3) + 2/(9*ArcCosh[2]^2) +
    1/(6*Sqrt[3]*ArcCosh[2])}

Two numerical solutions of the parameters {a,b,c} can be obtained

sol = NSolve[(0 == #) & /@ coeff, {a, b, c}, 36]

But the following coefficients in the Taylor expansion are not zero

N[(List @@
       Series[ArcCosh[2]/ArcCosh[2 - x] -
          Hypergeometric2F1[a, b, c, x], {x, 0, 7}][[3]])] /. sol //
   Simplify // Chop


{{0, 0, 0, -0.0004795487345876648, -0.0010026916113038686,
    -0.0014561420339331, -0.0018271816030282495},
   {0, 0, 0, -0.0004795487345877203, -0.0010026916113039241,
    -0.0014561420339331, -0.0018271816030282495}}

-- 

Roland Franzius


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