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MathGroup Archive 2006

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Re: Help finding x of hypergeometric 2F1[a,b,c,x] ?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71927] Re: Help finding x of hypergeometric 2F1[a,b,c,x] ?
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Mon, 4 Dec 2006 06:39:02 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <ekmet6$8mj$1@smc.vnet.net>

In article <ekmet6$8mj$1 at smc.vnet.net>, titus_piezas at yahoo.com wrote:

> Hello all,
> 
> The above function is given by Mathematica as
> Hypergeometric2F1[a,b,c,d].  Define the ff,
> 
> h1 = 2F1[a,b,1,1-x]
> h2 = 2F1[a,b,1,x]
> 
> Problem: Given a,b where 0<(a,b)<1, and a+b=1, find the unique real
> number x with 0<x<1 such that,
> 
>      h1/h2 = sqrt[n]
> 
> for arbitrary rational n>0.

I would define h1 and h2 as follows:

  h1[a_][x_] = Hypergeometric2F1[a, 1 - a, 1, 1 - x]

  h2[a_][x_] = Hypergeometric2F1[a, 1 - a, 1, x]

that is, as functions of one parameter a, and one variable x.
 
> For certain (a,b) namely, (1/2, 1/2), (1/3, 2/3), (1/4, 3/4) and (1/6,
> 5/6), closed-form solutions are known for x. For example, the first
> reduces to finding the "elliptic modulus k" and x can easily be given
> as x = ModularLambda[Sqrt[-n]].  Surprisingly though, it seems little
> is known for other (a,b).

Why surprisingly? New special functions, such as ModularLambda, are only 
introduced if they are frequently encountered.
> 
> Question: Is there Mathematica code to numerically evaluate x to
> arbitrary precision for any (a,b) and n? 

Yes -- and it is called FindRoot. For example, with a -> 1/2

  FindRoot[h1[1/2][x]/h2[1/2][x] == Sqrt[3/4], {x, 0.1},
     WorkingPrecision -> 50]

This gives the same answer as

  N[ModularLambda[Sqrt[-(3/4)]], 50]

> The fact that x is 0<x<1 greatly helps. 

Not sure why this "greatly helps". It is just a condition for the 
problem.

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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