A question concerning uniqueness JmsNxn Ultimate Fellow Posts: 1,214 Threads: 126 Joined: Dec 2010 10/05/2011, 04:28 PM (This post was last modified: 10/05/2011, 04:52 PM by JmsNxn.) My question is simple, and I hope somebody has an answer because I am a little confused. How come Tetration has multiple possible extensions to the complex domain that are analytic, but exponentiation only has one? Is it possible to have an alternative extension for exponentiation that is still analytic? It would have a piecewise definition, the gamma variable is to distinguish it from regular exponentiation: $a^{\otimes_\gamma\,\,k} = a\,\cdot\,a\,\cdot... a$ k amount of times if $k \in N$ and then, for \( 0eta, because for bases<=eta, the superfunction is real valued, so no Kneser mapping is necessary. For bases-2. - Sheldon Catullus Fellow Posts: 213 Threads: 47 Joined: Jun 2022 06/10/2022, 08:45 AM (This post was last modified: 07/12/2022, 05:01 AM by Catullus.) (10/05/2011, 04:28 PM)JmsNxn Wrote: My question is simple, and I hope somebody has an answer because I am a little confused.  How come Tetration has multiple possible extensions to the complex domain that are analytic, but exponentiation only has one?  Is it possible to have an alternative extension for exponentiation that is still analytic?Why, not how come. Why is better than how come. Regular exponentiation is not always analytic. For example, $0\uparrow x$ is not analytic at zero. Why a piecewise definition? Functions defined from piece-wise definitions are typically not analytic at the ends of pieces. Defining $a\uparrow\otimes_\gamma\uparrow b$ as $a\uparrow b*cos(b*\tau)$ might work. Defining $a\otimes_\gamma b$ as $a*b*cos(b*\tau)$ might work. Defining $a\oplus_\gamma b$ as $a+b+cos(b*\tau)$ might work. Though, these are not piece-wise definitions. Please remember to stay hydrated. ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\ « Next Oldest | Next Newest »

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