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 [split] Understanding Kneser Riemann method marraco Fellow Posts: 93 Threads: 11 Joined: Apr 2011 01/13/2016, 05:31 PM (This post was last modified: 01/13/2016, 05:38 PM by marraco.) (01/13/2016, 08:24 AM)andydude Wrote: @marraco, @tommy That certainly is a cool equation, even if it is easily provable. @everyone Also, I think I can express my earlier comment in different words now. Tetration is defined as the 1-initialized superfunction of exponentials. The previous functions discussed earilier are 3-initialized and 5-initialized, which makes them, not tetration, by definition. However, if there is an analytic continuation of the 1-initialized superfunction that overlaps with the 3-initialized superfunction, AND if on the overlap f(0) = 3, then they can be considered branches of the same function. But until that is proven, I don't think it's accurate to say that they're all "tetration". They are, however, iterated exponentials in the sense that they extend $\exp_b^n(3)$ to non-integer n. And so I would probably write these functions as $\exp_b^n(1),\, \exp_b^n(3),\, \exp_b^n(5)$ instead of saying that ${}^{n}b$ is a multivalued function that returns all three. That makes sense, but on other side, we need to solve equations like $\vspace{22}{^x \left(\sqrt[2]{2} \right)=3}$ (3>2, the asymptotic limit). Similarly, when we solve $\vspace{14}{e^{x}=-1}$ (-1<0, the asymptotic limit), we do not say that is is a function different than exponentiation. We just extend the domain to complex numbers. but this equation $\vspace{22}{^x \left(\sqrt[2]{2} \right)=3}$ has real solutions, unless we consider the pair $\vspace{16}{(x,\,^0a)}$ a new kind of number. If we use a new kind of number, then, for the main branch, $\vspace{16}{^0a=(1,1)}$; no more a real number. I have the result, but I do not yet know how to get it. « Next Oldest | Next Newest »

 Messages In This Thread [split] Understanding Kneser Riemann method - by andydude - 01/13/2016, 04:01 PM RE: Should tetration be a multivalued function? - by marraco - 01/13/2016, 05:31 PM RE: Should tetration be a multivalued function? - by sheldonison - 01/13/2016, 05:36 PM RE: Should tetration be a multivalued function? - by andydude - 01/13/2016, 06:41 PM RE: Should tetration be a multivalued function? - by andydude - 01/13/2016, 09:21 PM RE: Should tetration be a multivalued function? - by sheldonison - 01/13/2016, 10:55 PM RE: Should tetration be a multivalued function? - by andydude - 01/13/2016, 09:29 PM RE: Should tetration be a multivalued function? - by sheldonison - 01/13/2016, 10:58 PM

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