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Help finding x of hypergeometric 2F1[a,b,c,x] ?
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. 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). Question: Is there Mathematica code to numerically evaluate x to arbitrary precision for any (a,b) and n? The fact that x is 0<x<1 greatly helps. To illustrate, say for (1/5, 4/5) and given a certain n, one can find x up to a few digits precision by plugging in values x = 0.1, 0.2,...0.9 and observing how close (h1/h2)^2 comes to n, and, say, if it is between 0.4-0.5, then the values 0.41, 0.41,...0.50, and so on, with x becoming increasingly precise as the "range" where x lies becomes narrower. I did this manually (I know, crude) but there must be code for this. Any help will be appreciated. -Titus