Re: Hypergeometric integral looks wrong ?

*To*: mathgroup at smc.vnet.net*Subject*: [mg55500] Re: Hypergeometric integral looks wrong ?*From*: "luis serven" <junk1 at lafaena.com>*Date*: Sat, 26 Mar 2005 02:39:35 -0500 (EST)*References*: <d1tv31$rk0$1@smc.vnet.net> <d20r1q$baq$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Thanks for your reply. However, the transformation you mention involves real powers of (-z) ands is defined for z outside (0,1), while in my case I need to allow for 1> z >=0. Luigi "Maxim" <ab_def at prontomail.com> wrote in message news:d20r1q$baq$1 at smc.vnet.net... > On Thu, 24 Mar 2005 08:51:45 +0000 (UTC), luigi <junk1 at lafaena.com> wrote: > >> I have to compute the following integral: >> >> (1/ bv) Integrate[Exp[- b v s](1 - z Exp[-v s])^(-a), {s, 0, Infinity}, >> Assumptions -> {Re[b] > 0, Re[v] > 0, Re[b v] > 0}] >> >> In my case, z runs over (-Infinity,1), and a, b v are real ( a < 0). >> Now if the limits of integration were (0, Infinity) then this is just the >> hypergeometric function 2F1[a, b, b+1, z]. But since I am integrating >> over >> the interval (0, t), direct calculation yields >> >> 2F1[a, b, b+1, z] - Exp[-b v t] 2F1[a, b, b+1, z Exp [- v t]] >> >> But Mathematica 5.0 instead yields >> >> (1-z)^(-a) 2F1[a, a-b, 1+a-b , 1/z] ((z-1)/z)^(a) - Exp[-b v t] (1-Exp[v >> t]/z)^a (1 - Exp[-v t] z)^(-a) 2F1[a, a-b, 1+a-b , Exp[v t]/z ] >> >> This looks quite different and, moreover, does not seem to make sense for >> the range of z I am allowing. Is mathematica's result right ? Am I >> missing >> some transformation that links both results ? >> >> Thanks for any help. >> >> > > Yes, it is possible to convert one form of the answer to another, except > that b*v (and not bv) should be put in the numerator. Mathematica's answer > contains Hypergeometric2F1[..., E^(v*t)/z] rather than > Hypergeometric2F1[..., z*E^(-v*t)], so we need an appropriate > transformation identity. Using the formula found at > http://functions.wolfram.com/07.23.17.0057.01 , we obtain (in version 5.1) > > In[1]:= > Assuming[a < 0 && b > 0 && v > 0 && z < 1 && t > 0, > b*v*Integrate[E^(-b*v*s)*(1 - z*E^(-v*s))^(-a), {s, 0, t}] /. > Beta[z_, a_, b_] -> > z^a/a*Hypergeometric2F1[a, 1 - b, a + 1, z] /. > Hypergeometric2F1[a_, b_, c_, z_] -> > Gamma[b - a]*Gamma[c]/(Gamma[b]*Gamma[c - a])* > Hypergeometric2F1[a, a - c + 1, a - b + 1, 1/z]/(-z)^a + > Gamma[a - b]*Gamma[c]/(Gamma[a]*Gamma[c - b])* > Hypergeometric2F1[b, b - c + 1, b - a + 1, 1/z]/(-z)^b // > FullSimplify[#, ExcludedForms -> _Hypergeometric2F1]&] > > Out[1]= > Hypergeometric2F1[a, b, 1 + b, z] - > Hypergeometric2F1[a, b, 1 + b, z/E^(t*v)]/E^(b*t*v) > > as requested. > > Maxim Rytin > m.r at inbox.ru >