Re: Re: Hypergeometric2F1
- To: mathgroup at smc.vnet.net
- Subject: [mg93258] Re: [mg93156] Re: [mg93136] Hypergeometric2F1
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Sat, 1 Nov 2008 05:10:22 -0500 (EST)
- References: <200810280954.EAA22098@smc.vnet.net> <200810291049.FAA09463@smc.vnet.net> <490ADB4D.6060901@csl.pl> <8A64139D-4377-4DF0-BD6B-A61C7F9744D6@mimuw.edu.pl>
One more thing. When FindInstance returns {} it means that can prove
that there are no solutions to the problem. This is perfectly reliable
answer.
When FindInstance cannot decide, it tells you so.
Compare this:
FindInstance[x^4 + y^4 == -z^4 && x y z != 0, {x, y, z}, Integers]
{}
with this:
FindInstance[x^3 + y^3 == z^3 && x*y*z != 0, {x, y, z}, Integers]
The methods available to FindInstance are insufficient to find the
requested instances or prove they do not exist. >>
FindInstance[x^3 + y^3 == z^3 && x*y*z != 0, {x, y, z}, Integers]
(Besides learning some calculus it might be helpful to learn the
basics of Mathematica).
Andrzej Kozlowski
On 31 Oct 2008, at 20:21, Andrzej Kozlowski wrote:
> You seem to have missed some of you calculus classes, in particular,
> when the subject was Taylor's series (that is what + O[x]^5 in my
> code means).
>
> (As form Fermat's Last Theorem; actually Mathematica knows it:
>
> FullSimplify[x^n + y^n == z^n, Element[x | y | z | n, Integers] && n
> > 2 && x y z != 0]
> False
> )
>
>
> Andrzej Kozlowski
>
> On 31 Oct 2008, at 19:17, Artur wrote:
>
>> I'm agree that my previous sample wasn't good ArcCosh[2]/ArcCosh[2-
>> x] with FindInstance (I was think about FindFit (interpolating).
>> The best interpolation known for me is:
>> Plot[{ArcCosh[2]/ArcCosh[2 - x], Hypergeometric2F1[1/2, 2/3, 4/5,
>> x]}, {x, -4, 1}]
>> I'm looking for better
>>
>> I'm still looking for working procedure procedure for:
>> FindInstance[Hypergeometric2F1[1/3, a/3, 5/6, b/32] == 8/5, {a, b}]
>> what inspite FindInstance ?
>>
>> or
>>
>> FindInstance[2 Cos[2 Pi x] == Hypergeometric2F1[1/2 + a x, 1/2 + b
>> x, 1/2, 3/4] ,{a,b}]
>>
>> As I was informed earlier procedure of Andrzej Kozlowski don't work
>> with Hypergeometric2F1
>> e.g.
>> In[1]: FindInstance[ LogicalExpand[
>> 2 Cos[2 Pi x] - Hypergeometric2F1[1/2 + a x, 1/2 + b x, 1/2, 3/4] +
>> O[x]^5 == 0], {a, b}]
>> Out[1]:{}
>>
>> True answer is {a,b}={-3,3} or {a,b}={3,-3}
>>
>> Because always Andrzej Kozlowski's procedure return {} if is used
>> with Hypergeometric2F1 we can also prooved with use of them Fermat
>> Last Theory.
>>
>> Between beliving that procedure work and true working is infinity.
>>
>> Best wishes
>> Artur
>>
>> Andrzej Kozlowski pisze:
>>> On 28 Oct 2008, at 18:54, Artur wrote:
>>>
>>>
>>>> Dear Mathematica Gurus!
>>>> Who know which Mathematica procedure to use to find such a,b,c that
>>>> ArcCosh[2]/ArcCosh[2-x]==Hypergeometric2F1[a,b,c,x] for {x,-
>>>> Infinity, 1}
>>>> BEST WISHES
>>>> ARTUR
>>>>
>>>>
>>>
>>>
>>> What makes you think such a,b,c exist?
>>> This seems to indicate that they do not:
>>>
>>> FindInstance[LogicalExpand[
>>> ArcCosh[2]/ArcCosh[2 - x] -
>>> Hypergeometric2F1[a, b, c, x] + O[x]^5 ==
>>> 0], {a, b, c}]
>>> {}
>>>
>>> Andrzej Kozlowski
>>>
>>>
>>>
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>