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