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 Could be tetration if this integral converges JmsNxn Long Time Fellow Posts: 571 Threads: 95 Joined: Dec 2010 05/06/2014, 12:11 AM (This post was last modified: 05/06/2014, 12:19 AM by JmsNxn.) (05/05/2014, 11:45 PM)mike3 Wrote: (05/05/2014, 04:27 PM)JmsNxn Wrote: (05/05/2014, 02:11 AM)mike3 Wrote: (05/04/2014, 09:06 PM)JmsNxn Wrote: $F(n) = \sum_{j=0}^n \frac{n!(-\lambda)^{n-j}}{j!(n-j)!(^j e)}$ So is this $F$ supposed to approximate tetration if $\lambda$ is small? As if so, then it doesn't seem to be working for me. If I take $\lambda = 0.01$ and the integral upper bound at 2000, I get $F(1.5)$ as ~443444.33873479713260158296678612894384. Clearly, that can't be right -- it should be between $e$ and $e^e$ (if this is supposed to reproduce the Kneser tetrational then it should be ~5.1880309584291901006085359610758671512). It gets worse the smaller you make $\lambda$ -- i.e. it doesn't seem to converge. Also, picking values to put in that are near-natural numbers doesn't seem to work, either.I'll note firstly that $F(n) = \sum_{j=0}^n \frac{n!\lambda^{n-j}}{j!(n-j)!(^j e)}$ I accidentally added an extra negative. But that doesn't really affect convergence. I understand whats happening. Hmm. That makes sense now that I think about it. I was hoping you could take lambda small but not too small and it wouldn't diverge too fast but because obviously $\lambda = 0$ diverges this doesn't happen. Maybe if you try $\lambda = 1$ I've done more research into this form of the operator so perhaps we can work with this one. $F(n) = \sum_{j=0}^n \frac{(-1)^{n-j} n!}{j!(n-j)!(^j e)}$ Well I tried this $F$ and the integral also didn't seem to converge. Trying $\lambda = 1$ yields a finite value but the recurrence $F(z+1) = \exp(F(z))$ does not appear to hold, nor are the values close to those of the Kneser tetrational. I noticed your discussion after this point about the continuum sum thing and you mentioned about fractional iteration of the difference operator. This is why I was curious as to how the integral definition for the Weyl differintegral related to its definition for periodic functions. In particular, if $f(z)$ is periodic with period $2\pi$, we have a Fourier series $f(z) = \sum_{n=-\infty}^{\infty} a_n e^{inz)$. We assume $a_0 = 0$. Then, $D^s f(z) = \sum_{n=-\infty}^{\infty} a_n (in)^s e^{inz}$. This is the Weyl differintegral. Note that taking $s = -1$ yields the integral (though we have to drop the term at $n = 0$). Similarly, for the finite difference operator, $\Delta^s f(z) = \sum_{n=-\infty}^{\infty} a_n (e^{in} - 1)^s e^{inz}$. Note that taking $s = -1$ yields the continuum sum (though, again, we have to drop the term at $n = 0$). Now, if the first expression for the differintegral can be generalized to certain non-periodic holomorphic functions via an integral transform, can that also be done for the second? Is there a method to derive the integral transform from the given definition in the first case? If so, can it be generalized to the second? If I understand your question correctly the answer is yes to both questions. I'll write it out. Using the operators from my paper we can completely represent the iterated difference: $F(z) =[\frac{d^{-z}}{dx^{-z}}e^x \beta(-x)]_{x=0}$ then: $\bigtriangledown^s F(z) = [\frac{d^{-z}}{dx^{-z}}e^x \frac{d^s}{d(-x)^s}\beta(-x)]_{x=0}$ I'm working on writing this all up. So far all I have is a bunch of notes and papers compiled together unorganized. Now for the first question, to work on these periodic functions define: $[\frac{d^z}{dw^z} f(w)]_{w=0} = \frac{1}{\G(-z)} (\sum_{n=0} f^{(n)}(0) \frac{(-1)^n}n!(n-z)} + \int_1^\infty f(-x)x^{-z-1}\,dx)$ And then we can generate the differintegral using taylor series. now if $p(w) = \sum_{n=1} a_n e^{inw}$ then $\frac{d^{-z}}{dw^{-z}} p(w) = \frac{i^{-z}}{\G(z)} \int_0^\infty p(w + ix)x^{z-1}\,dx$ « Next Oldest | Next Newest »

 Messages In This Thread Could be tetration if this integral converges - by JmsNxn - 04/03/2014, 02:14 PM RE: Could be tetration if this integral converges - by sheldonison - 04/30/2014, 11:17 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/02/2014, 03:33 PM RE: Could be tetration if this integral converges - by tommy1729 - 04/30/2014, 12:29 PM RE: Could be tetration if this integral converges - by mike3 - 05/03/2014, 01:19 AM RE: Could be tetration if this integral converges - by tommy1729 - 05/11/2014, 04:26 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/11/2014, 04:30 PM RE: Could be tetration if this integral converges - by tommy1729 - 05/11/2014, 04:52 PM RE: Could be tetration if this integral converges - by mike3 - 05/03/2014, 05:24 AM RE: Could be tetration if this integral converges - by mike3 - 05/03/2014, 07:13 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/03/2014, 06:12 PM RE: Could be tetration if this integral converges - by mike3 - 05/04/2014, 02:18 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/04/2014, 07:27 PM RE: Could be tetration if this integral converges - by mike3 - 05/05/2014, 12:55 AM RE: Could be tetration if this integral converges - by mike3 - 05/04/2014, 11:50 AM RE: Could be tetration if this integral converges - by sheldonison - 05/04/2014, 03:28 PM RE: Could be tetration if this integral converges - by mike3 - 05/05/2014, 01:00 AM RE: Could be tetration if this integral converges - by sheldonison - 05/05/2014, 03:49 PM RE: Could be tetration if this integral converges - by tommy1729 - 05/04/2014, 01:25 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/04/2014, 07:36 PM RE: Could be tetration if this integral converges - by MphLee - 05/04/2014, 07:44 PM RE: Could be tetration if this integral converges - by mike3 - 05/04/2014, 10:42 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/04/2014, 11:32 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/04/2014, 09:06 PM RE: Could be tetration if this integral converges - by mike3 - 05/05/2014, 02:11 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/05/2014, 04:27 PM RE: Could be tetration if this integral converges - by mike3 - 05/05/2014, 11:45 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/06/2014, 12:11 AM RE: Could be tetration if this integral converges - by mike3 - 05/06/2014, 06:50 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/06/2014, 03:54 PM RE: Could be tetration if this integral converges - by mike3 - 05/07/2014, 03:25 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/07/2014, 03:18 PM RE: Could be tetration if this integral converges - by mike3 - 05/11/2014, 07:47 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/11/2014, 04:29 PM RE: Could be tetration if this integral converges - by mike3 - 05/11/2014, 11:26 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/12/2014, 01:44 AM RE: Could be tetration if this integral converges - by mike3 - 05/12/2014, 02:15 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/12/2014, 03:32 PM RE: Could be tetration if this integral converges - by tommy1729 - 05/12/2014, 11:26 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/13/2014, 01:58 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/05/2014, 06:18 PM RE: Could be tetration if this integral converges - by tommy1729 - 05/05/2014, 09:09 PM

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