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 Rational operators (a {t} b); a,b > e solved JmsNxn Long Time Fellow Posts: 566 Threads: 94 Joined: Dec 2010 06/06/2011, 08:47 AM (This post was last modified: 06/06/2011, 09:01 AM by JmsNxn.) (06/06/2011, 06:53 AM)bo198214 Wrote: But James, this is not analytic at $t=1$, if we reformulate: $ f(t) = a\, \{t\}\, b = \left\{ \begin{array}{c l} \exp_\eta^{\circ t}(\exp_\eta^{\circ-t}(a) + \exp_\eta^{\circ -t}(b)) & t \in (-\infty,1]\\ \exp_\eta^{\circ t}(\exp_\eta^{\circ -t}(a)+\exp_\eta^{\circ -1}(b)) & t \in [1,2] \end{array} \right.$ We can say: $a\, \{t\}\, b = \exp_\eta^{\circ t}(\exp_\eta^{\circ -t}(a) + h_b(t))$ where $h_b(t)=\left{\begin{array}{c l} \exp_\eta^{\circ -t}(b) & t\le 1\\ \exp_\eta^{\circ -1}(b) & t\in [1,2] \end{array}\right.$ $f$ is addition and composition of analytic functions, except this one function $h_b$. The whole function $f(t)$ can not be analytic. I wonder why it looks so smooth. I like your definition better--it seems sleeker . I was sort of aware that there was no way I was gonna produce an analytic function over the whole complex domain, I'm happy with analytic in a few regions. Quote:But I see you gracefully avoided that problem by just defining it for a,b > e well hopefully I'll be having to tackle that problem soon. Quote:PS: 1. $g(t) = a\, \} t \{ \, b$, This notation is ambiguous, compare $\{ a \} t \{ b \} + c$. Please invent a better one! Alright, from henceforth I shall refer to logarithmic semi operators with the following notation: $a\, \bigtriangleup_t\, b = a \,\{t\}\,b$ And the inverse is given by: $a\, \bigtriangledown_t\, b = a \,\}t\{\, b$ therefore: $a \bigtriangleup_0 b = a + b\\ a \bigtriangleup_1 b = a * b\\ a \bigtriangledown_0 b = a - b$ etc etc.. Quote:2. $\exp_\eta^{\alpha t}$, not \alpha but \circ belongs in the exponent: $\exp_\eta^{\circ t}$. This notation is derived from the symbol for function composition $f\circ g$. I knew there was something off about my equations. lol (06/06/2011, 06:02 AM)sheldonison Wrote: Hey James, try my code snippet, which I updated while you were posting. It will work for values of a and b 2... at least I think so. « 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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