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 Is the following expression for tetration useful? mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 11/07/2012, 09:19 PM (This post was last modified: 11/07/2012, 10:48 PM by mike3.) (11/07/2012, 08:23 PM)JmsNxn Wrote: I've been doing a lot of research in areas around linear operators and I've found the following theorem. If it's useful I'll prove how I got it. If not I won't. We can express the following: $\bf{E}f(s) = f(e^s) = \frac{1}{\pi} \int_1^{\infty} f(-u) \int_{-\infty}^{\infty} sin{\pi t}\,\cdot\, e^{-\pi i t}\,\cdot\,\frac{s^t}{t^2}\,\cdot\,\ln(u)^{-t}\,\partial t \partial u$ $\bf{E}^{-1}f(s) = f(\ln(s)) = \frac{-1}{\pi} \int_{-\infty}^{\infty} f(u) \int_{-\infty}^{\infty} sin{\pi t}\,\cdot\, e^{(u-\pi i )t}\,\cdot\,\frac{s^t}{t^2}\,\cdot\,\partial t \partial u$ And so therefore we can write: $\exp^{\circ\,y}(s) = \bf{E}^{y-1} e^s$ If you can't notice $\bf{E}$ is a linear operator; so: $\bf{E}(\alpha f + \beta g) = \alpha \bf{E} f + \beta \bf{E} g$ This result is quite elaborate to prove and requires knowledge of Hilbert spaces. I just found this expression recently of a more general result that I am more interested in. We must remember this is right hand composition. Nonetheless; is it easier to iterate a linear operator than how we usually do it? I can apply these methods for pentation and every hyper operator; formally; without considering convergence of the integrals. I'm still in the baby steps. Questions, comments? I can do everything I just did for iteration of any base as well. Not sure about convergence though. We can actually turn every super-function into iteration of a linear transformation. However; again; formally; not sure about convergence. I'm working on a paper that proves all of this but feedback helps; maybe someone's seen this. Edit: What's cool about this is we do not require a fixpoint! Interesting. Could you give a few more details about how you got it -- e.g. what other theorems did you use, etc.? I just posted a question to mathoverflow here: http://mathoverflow.net/questions/111752...-transform (EDIT: I Deleted this -- see NEW POST) « Next Oldest | Next Newest »

 Messages In This Thread Is the following expression for tetration useful? - by JmsNxn - 11/07/2012, 08:23 PM RE: Is the following expression for tetration useful? - by mike3 - 11/07/2012, 09:19 PM RE: Is the following expression for tetration useful? - by JmsNxn - 11/07/2012, 10:11 PM RE: Is the following expression for tetration useful? - by mike3 - 11/07/2012, 10:38 PM RE: Is the following expression for tetration useful? - by mike3 - 11/07/2012, 10:52 PM RE: Is the following expression for tetration useful? - by JmsNxn - 11/08/2012, 02:40 PM RE: Is the following expression for tetration useful? - by mike3 - 11/10/2012, 03:19 AM RE: Is the following expression for tetration useful? - by tommy1729 - 11/13/2012, 11:50 PM RE: Is the following expression for tetration useful? - by tommy1729 - 11/12/2012, 04:12 PM RE: Is the following expression for tetration useful? - by tommy1729 - 11/14/2012, 11:29 PM

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