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 >>> >>> >>> >>> __________ Information from ESET NOD32 Antivirus, version of virus >>> signature database 3565 (20081029) __________ >>> >>> The message was checked by ESET NOD32 Antivirus. >>> >>> http://www.eset.com >>> >>> >>> >>> >