Re: Re: Annoying error in Hypergeometric2F1
- To: mathgroup at smc.vnet.net
- Subject: [mg107928] Re: [mg107902] Re: Annoying error in Hypergeometric2F1
- From: anguzman at ing.uchile.cl
- Date: Wed, 3 Mar 2010 05:50:23 -0500 (EST)
- References: <hmikv0$55f$1@smc.vnet.net>
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 at uos.de> ha escrito:
> anguzman at 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@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
>
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