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 Rational operators (a {t} b); a,b > e solved JmsNxn Long Time Fellow Posts: 739 Threads: 104 Joined: Dec 2010 06/06/2011, 07:47 PM (This post was last modified: 06/06/2011, 07:58 PM by JmsNxn.) Alright, testing the left hand right hand limit I get different values.... I'll refer to $\vartheta(a, b, \sigma) = a\,\,\bigtriangle_\sigma\,\,b$ $\{a,b, \sigma| \R(a), \R(b) > e\,;\,a,b,\sigma \in C\}$ from now on. So therefore: $\lim_{h\to 0^+} \frac{\large \vartheta(3, 4, 1 + h) - \vartheta(3, 4, 1)}{h} = 10.7633\\ \lim_{h\to 0^-} \frac{\large \vartheta(3, 4, 1 + h) - \vartheta(3, 4, 1)}{h} = 9.9206$ and $\lim_{h\to 0^+} \frac{\large \vartheta(3, 3, 1 + h) - \vartheta(3, 3, 1)}{h} = 5.4105\\ \lim_{h\to 0^-} \frac{\large \vartheta(3, 3, 1 + h) - \vartheta(3, 3, 1)}{h} = 5.3674$ And finally, I'd like to post a question for anyone more familiar with iteration dynamics than me. This is the open problem to extend $\R(\sigma) \in [2, 3]$ if $f(z) = a \,\,\bigtriangleup_{1+q}\,\, z = \vartheta(a, z, 1+q)$ then: $\vartheta(a, t, 2+q) = f^{\circ t}(1)$ and so solving for [2, 3] is solving for the iterate of f(1). f(z) is an analytic function since it's composed of analytic functions and q is restricted to [0,1]. hmmm, so the values come close to each other but don't quite make it. I wonder, does this disqualify it considering it's not analytic at t=1 and the derivative is undefined? The only case where it is analytic and it is defined is $\vartheta(e, e, \sigma) = \text{cheta}(\sigma)\,\,\,\, \R(\sigma) \le 2$ Which of course I tested using code over the exponential period [1, 2]. i.e.: $0 \le q \le 1$ $\lim_{h\to 0}\, \vartheta(e+h, e+h, 1 + q)\, =\, \text{cheta}(1+q)$ But I think that makes it even more beautiful, the fact that it's only analytic at 1 when a, b = e And finally, I'd like to post a question for anyone more familiar with iteration dynamics than me. This is the open problem to extend $\R(\sigma) \in [2, 3]$ if $f(z) = a \,\,\bigtriangleup_{1+q}\,\, z = \vartheta(a, z, 1+q)$ then: $\vartheta(a, t, 2+q) = f^{\circ t}(1)$ and so solving for [2, 3] is solving for the iterate of f(1). f(z) is an analytic function since it's composed of analytic functions. « Next Oldest | Next Newest »

 Messages In This Thread Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/06/2011, 02:45 AM RE: Rational operators (a {t} b); a,b > e solved - by sheldonison - 06/06/2011, 04:39 AM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/06/2011, 05:34 AM RE: Rational operators (a {t} b); a,b > e solved - by sheldonison - 06/06/2011, 06:02 AM RE: Rational operators (a {t} b); a,b > e solved - by sheldonison - 06/06/2011, 07:03 AM RE: Rational operators (a {t} b); a,b > e solved - by nuninho1980 - 06/06/2011, 05:16 PM RE: Rational operators (a {t} b); a,b > e solved - by bo198214 - 06/06/2011, 06:53 AM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/06/2011, 08:47 AM RE: Rational operators (a {t} b); a,b > e solved - by bo198214 - 06/06/2011, 09:23 AM RE: Rational operators (a {t} b); a,b > e solved - by tommy1729 - 06/06/2011, 11:59 AM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/06/2011, 05:44 PM RE: Rational operators (a {t} b); a,b > e solved - by bo198214 - 06/06/2011, 09:28 PM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/06/2011, 07:47 PM RE: Rational operators (a {t} b); a,b > e solved - by sheldonison - 06/06/2011, 08:43 PM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/07/2011, 02:45 AM RE: Rational operators (a {t} b); a,b > e solved - by bo198214 - 06/07/2011, 06:59 AM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/08/2011, 04:54 AM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/08/2011, 07:31 PM RE: Rational operators (a {t} b); a,b > e solved - by sheldonison - 06/08/2011, 08:32 PM RE: Rational operators (a {t} b); a,b > e solved - by bo198214 - 06/08/2011, 09:14 PM RE: Rational operators (a {t} b); a,b > e solved - by tommy1729 - 09/02/2016, 01:50 AM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/08/2011, 11:47 PM RE: Rational operators (a {t} b); a,b > e solved - by Gottfried - 06/11/2011, 02:33 PM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/12/2011, 07:55 PM RE: Rational operators (a {t} b); a,b > e solved - by Xorter - 08/21/2016, 06:56 PM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 08/22/2016, 12:36 AM RE: Rational operators (a {t} b); a,b > e solved - by Xorter - 08/24/2016, 07:24 PM RE: Rational operators (a {t} b); a,b > e solved - by Xorter - 08/29/2016, 02:06 PM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 09/01/2016, 06:47 PM RE: Rational operators (a {t} b); a,b > e solved - by tommy1729 - 09/02/2016, 02:04 AM RE: Rational operators (a {t} b); a,b > e solved - by tommy1729 - 09/02/2016, 02:11 AM

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