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 Taylor series of upx function Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 11/17/2008, 04:11 AM sheldonison Wrote:... To me, it would seem as if the first couple of derivatives at x=-1,0 were the same for both solutions, than it is likely that they're the same solution, though the two could still diverge in the higher derivatives. I had submitted the evaluation of the first derivative with 14 (I hope) decimal digits. $\mathrm{tet}'(-1)=\mathrm{tet}'(0)\approx 1.09176735125832$ In the similar way, the second derivative can be evaluated, and so on. But I like the approach by Bo: it is better to prove, than to compare the numerical evaluations. We already have proof that no other tetration is allowed to be holomorphic in the complex plane with cutted out set $\{x~\in~ \mathbb{R} ~:~x\le -2 \}$. By the way, how about to add some small amount of finction $d(z)= \left\{ \begin{array}{ccc} \exp\!\Big(-0.1/x^2 - 0.1/(z+1)^2\big) &,& (x+1)x \ne 0\\ 0 &,& (x+1)x = 0 \end{array}\right.$?THis function has all the derivatives along the real axis, and all of them are zero at points 0 and -1. Consider the modification of tetration $\mathrm{tet}(z)$ to $\mathrm{tem}(z)=\mathrm{tet}(z)+d(z)$ with corresponding extension from the range [-1,0] usind the recurrent equation. At the real axis, the modified tetration tem has the same "expansion" as the tetration in these points; however, such a modification destroys the holomorphism. « Next Oldest | Next Newest »

 Messages In This Thread Taylor series of upx function - by Zagreus - 09/07/2008, 10:56 AM RE: Taylor series of upx function - by bo198214 - 09/10/2008, 01:56 PM RE: Taylor series of upx function - by andydude - 10/23/2008, 10:16 PM RE: Taylor series of upx function - by Kouznetsov - 11/16/2008, 01:40 PM RE: Taylor series of upx function - by bo198214 - 11/16/2008, 07:17 PM RE: Taylor series of upx function - by sheldonison - 11/17/2008, 12:21 AM RE: Taylor series of upx function - by Kouznetsov - 11/17/2008, 04:11 AM RE: Taylor series of upx function - by bo198214 - 11/18/2008, 11:49 AM RE: Taylor series of upx function - by sheldonison - 03/05/2009, 06:48 PM RE: Taylor series of upx function - by Kouznetsov - 12/02/2008, 12:03 PM RE: Taylor series of upx function - by sheldonison - 12/03/2008, 09:11 PM RE: Taylor series of upx function - by Kouznetsov - 12/05/2008, 12:30 AM

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