Annoying error in Hypergeometric2F1
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From: anguzman on 2 Mar 2010 04:15 This is very bad and disappointing...what about version 7 .. Looks like the symbolic evaluation is messed up.. Mathematica 6.0 for Linux x86 (32-bit) Copyright 1988-2008 Wolfram Research, Inc. In[1]:= Hypergeometric2F1[1/2, 2, 3/2, -125/100] Sqrt[5] 10 + 9 Sqrt[5] ArcTanh[-------] 2 Out[1]= ------------------------------- 45 In[2]:= %//N Out[2]= 0.867836 - 0.702481 I In[3]:= Hypergeometric2F1[1/2, 2, 3/2, -1.25] Out[3]= 0.59836 Atte. Andres Guzman ---------------------------------------------------------------- This message was sent using IMP, the Internet Messaging Program.
From: Roland Franzius on 2 Mar 2010 07:57 anguzman(a)ing.uchile.cl schrieb: > This is very bad and disappointing...what about version 7 .. > Looks like the symbolic evaluation is messed up.. > > Mathematica 6.0 for Linux x86 (32-bit) > Copyright 1988-2008 Wolfram Research, Inc. > > In[1]:= Hypergeometric2F1[1/2, 2, 3/2, -125/100] > > Sqrt[5] > 10 + 9 Sqrt[5] ArcTanh[-------] > 2 > Out[1]= ------------------------------- > 45 > > In[2]:= %//N > > Out[2]= 0.867836 - 0.702481 I > > In[3]:= Hypergeometric2F1[1/2, 2, 3/2, -1.25] > > Out[3]= 0.59836 > The Hypergeometric 2F1(a,b,c,z) has logarithmic branch points at z=+-1. That follows immediately using your expression by In: TrigToExp[ArcTanh[x]] Out: -(1/2) Log[1 - x] + 1/2 Log[1 + x] Without assumptions you cannot use it for real |z|>1. Probably you even don't know the actual meaning of the expression of yours. Consider In: Assuming[{x, y} > 1, FullSimplify(a)Hypergeometric2F1[1/2, 2, 3/2, x + I y]] Out: 1/2 (1/(1 - x - I y) + ArcTanh[Sqrt[x + I y]]/Sqrt[x + I y]) In: % /. {x -> -125/100, y -> 0} // N Out: 0.59836 -- Roland Franzius
From: Bob Hanlon on 2 Mar 2010 07:58 $Version 7.0 for Mac OS X x86 (64-bit) (February 19, 2009) Hypergeometric2F1[1/2, 2, 3/2, -125/100] (1/45)*(10 + 9*Sqrt[5]* ArcTan[Sqrt[5]/2]) % // N 0.59836 Hypergeometric2F1[1/2, 2, 3/2, -1.25] 0.59836 Hypergeometric2F1[1/2, 2, 3/2, -125/100] - Hypergeometric2F1[1/2, 2, 3/2, -1.25] -1.1102230246251565*^-16 Bob Hanlon ---- anguzman(a)ing.uchile.cl wrote: ============= This is very bad and disappointing...what about version 7 .. Looks like the symbolic evaluation is messed up.. Mathematica 6.0 for Linux x86 (32-bit) Copyright 1988-2008 Wolfram Research, Inc. In[1]:= Hypergeometric2F1[1/2, 2, 3/2, -125/100] Sqrt[5] 10 + 9 Sqrt[5] ArcTanh[-------] 2 Out[1]= ------------------------------- 45 In[2]:= %//N Out[2]= 0.867836 - 0.702481 I In[3]:= Hypergeometric2F1[1/2, 2, 3/2, -1.25] Out[3]= 0.59836 Atte. Andres Guzman
From: Christoph Lhotka on 2 Mar 2010 08:02 sorry, but neither with V5, V6 or V7 I could reproduce your result. I get: In[1]:= Hypergeometric2F1[1/2, 2, 3/2, -125/100] Sqrt[5] 10 + 9 Sqrt[5] ArcTan[-------] 2 Out[1]= ------------------------------- 45 In[2]:= %//N Out[2]= 0.59836 In[3]:= Hypergeometric2F1[1/2, 2, 3/2, -1.25] Out[3]= 0.59836 (the ArcTan instead of the ArcTanh) Another question: I remember a definition, where 2F1(a,b,c,x) is defined only for |x|<1. Does anybody know why / about the generalization? chr anguzman(a)ing.uchile.cl wrote: > This is very bad and disappointing...what about version 7 .. > Looks like the symbolic evaluation is messed up.. > > Mathematica 6.0 for Linux x86 (32-bit) > Copyright 1988-2008 Wolfram Research, Inc. > > In[1]:= Hypergeometric2F1[1/2, 2, 3/2, -125/100] > > Sqrt[5] > 10 + 9 Sqrt[5] ArcTanh[-------] > 2 > Out[1]= ------------------------------- > 45 > > In[2]:= %//N > > Out[2]= 0.867836 - 0.702481 I > > In[3]:= Hypergeometric2F1[1/2, 2, 3/2, -1.25] > > Out[3]= 0.59836 > > > > Atte. Andres Guzman > > ---------------------------------------------------------------- > This message was sent using IMP, the Internet Messaging Program. > > > > > -- Universit�t Wien || University of Vienna Institut f�r Astronomie || Institute for Astronomy T�rkenschanzstra�e 17 || Tuerkenschanzstrasse 17 1180 Wien || A-1180 Vienna 01 4277 51841 || 0043 1 4277 51841
From: anguzman on 3 Mar 2010 05:50
I have seen now that Mathematica 7 gives the correct result. Well, I found the inconsistency while I was trying to evaluate: In = Assuming[A > 0 && B > 0 && b > 0, Integrate[(A^2 + x^2)^(b), {x, 0, B}]] Out = A^(-2 b) B Hypergeometric2F1[1/2, b, 3/2, -(B^2/A^2)] An integral of real positive argument, the problem I exposed occurs with b=2. If there are branch discontinuities, you have to know which branch is selected in the evaluation though. But the problem is that there needs to be an ArcTan instead of a ArcTanh. Atte. Andres Guzman Roland Franzius <roland.franzius(a)uos.de> ha escrito: > anguzman(a)ing.uchile.cl schrieb: >> This is very bad and disappointing...what about version 7 .. >> Looks like the symbolic evaluation is messed up.. >> >> Mathematica 6.0 for Linux x86 (32-bit) >> Copyright 1988-2008 Wolfram Research, Inc. >> >> In[1]:= Hypergeometric2F1[1/2, 2, 3/2, -125/100] >> >> Sqrt[5] >> 10 + 9 Sqrt[5] ArcTanh[-------] >> 2 >> Out[1]= ------------------------------- >> 45 >> >> In[2]:= %//N >> >> Out[2]= 0.867836 - 0.702481 I >> >> In[3]:= Hypergeometric2F1[1/2, 2, 3/2, -1.25] >> >> Out[3]= 0.59836 >> > > The Hypergeometric 2F1(a,b,c,z) has logarithmic branch points at z=+-1. > That follows immediately using your expression by > > In: TrigToExp[ArcTanh[x]] > Out: -(1/2) Log[1 - x] + 1/2 Log[1 + x] > > Without assumptions you cannot use it for real |z|>1. > Probably you even don't know the actual meaning of the expression of yours. > > Consider > > In: Assuming[{x, y} > 1, > FullSimplify(a)Hypergeometric2F1[1/2, 2, 3/2, x + I y]] > > > Out: 1/2 (1/(1 - x - I y) + ArcTanh[Sqrt[x + I y]]/Sqrt[x + I y]) > > In: % /. {x -> -125/100, y -> 0} // N > > Out: 0.59836 > > -- > > Roland Franzius > > > > > > > ---------------------------------------------------------------- This message was sent using IMP, the Internet Messaging Program. |