*To*: mathgroup@smc.vnet.net*Subject*: [mg12160] Re: [mg12097] Boundary cond. at Infinity*From*: Bob Hanlon <BobHanlon@aol.com>*Date*: Mon, 27 Apr 1998 01:46:44 -0400

f[x_, a_] := Hypergeometric2F1[(a + 1)/2, 1, (a + 1)/2 + 1, -Exp[2*x]] There is a discontinuity at Solve[(a + 1)/2 + 1 == 0, a] {{a -> -3}} Plot3D[Hypergeometric2F1[(a + 1)/2, 1, (a + 1)/2 + 1, -Exp[2*x]], {a, -4, 4}, {x, 0, 4}, PlotPoints -> 21, AxesLabel -> {"a", "x", None}]; Linear transformation (Abramowitz and Stegun, 15.3.5): trans = Hypergeometric2F1[a_, b_, c_, z_] -> Hypergeometric2F1[b, c - a, c, z/(z - 1)]/(1 - z)^b; Simplify[f[x, a] /. trans] Hypergeometric2F1[1, 1, (a + 3)/2, Exp[2*x]/(1 + Exp[2*x])]/ (1 + Exp[2*x]) As x approaches Infinity, the numerator will tend towards Hypergeometric2F1[1, 1, (a + 3)/2, 1] (a + 1)/(a - 1) And the denominator tends to Infinity. The limit is then zero. Bob Hanlon In a message dated 4/25/98 6:18:19 AM, gold@dresden.isi.kfa-juelich.de wrote: >And Math.3.0 give me an answer in terms of PolyGamma's and >Hypergeometric2F1's with one constant C[2] which I want to find by >myself. To find it I need to know the behavior of e.g. >Hypergeometric2F1[(1 + a)/2, 1, 1 + (1 + a)/2, -E^(2*x)] at infinity. >How can I ask Mathematica to show me the behavior of arbitrary function >at infinity? I expect to get something like x*Exp[-2*x] i.e. Taylor is >of no use in this case since the behavior is exponential with unknown >power.