• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 exponential distributivity JmsNxn Long Time Fellow Posts: 464 Threads: 85 Joined: Dec 2010 09/22/2011, 03:27 PM (This post was last modified: 09/22/2011, 03:34 PM by JmsNxn.) (02/23/2008, 12:29 AM)quickfur Wrote: That's interesting, because analysing f(x+y) = f(x)f(y) leads to the exponential function $e^x$. Unfortunately, we don't seem to have such a handy property involving tetration that we can use, except for $f(x+1)=b^{f(x)}$, which isn't very helpful for non-integer x. Now that I think of it, I think the root of the problem is that exponentiation is non-associative, so there's no easy way to algebraically "access the top of the exponential tower", so to speak. In order to derive any useful relations, we need to have some way of "reaching the top of the tower". For example, given b[4]n, if we can somehow reach the top of the tower and add another tower of height m, then we can state the property that J(b[4]n, b[4]m) = b[4](n+m), where the hypothetical J operator attaches the second tower to the top of the first tower. But I doubt that J is expressible as an algebraic operation (I'm not even sure if it can be consistently defined if b is not fixed!). And so even if we could make such a statement, it wouldn't be of the same utility as $f(x+1)=b^{f(x)}$ w.r.t. exponentiation. I'd like to respectfully disagree. consider: $\mu (x) =\, ^x \mu$ $a \,\otimes_\mu\,b = \mu(\mu^{-1}(a) + \mu^{-1}(b))$ it's very easy to see that: $^c \mu \, \otimes_\mu\,^d \mu = \,^{c+d} \mu$ I think even, this operator may give some short cuts for algebra involving tetration. I did a little research into this under the following thread: http://math.eretrandre.org/tetrationforu...hp?tid=699 The only rule is if we create: $\phi (x) = \,^x \phi$ $a \,\otimes_\phi \,b = \phi(\phi^{-1}(a) + \phi^{-1}(b))$ $\otimes_\mu \neq \otimes_\phi$ whereas when we create the same operator with exponentiation instead of tetration, the operators are equivalent and both are multiplication. « Next Oldest | Next Newest »

 Messages In This Thread exponential distributivity - by bo198214 - 02/22/2008, 12:36 PM RE: exponential distributivity - by quickfur - 02/22/2008, 06:51 PM RE: exponential distributivity - by bo198214 - 02/22/2008, 07:17 PM RE: exponential distributivity - by quickfur - 02/23/2008, 12:29 AM RE: exponential distributivity - by JmsNxn - 09/22/2011, 03:27 PM

 Possibly Related Threads... Thread Author Replies Views Last Post Math overflow question on fractional exponential iterations sheldonison 4 8,037 04/01/2018, 03:09 AM Last Post: JmsNxn An exponential "times" table MikeSmith 0 2,840 01/31/2014, 08:05 PM Last Post: MikeSmith exponential baby Mandelbrots? sheldonison 0 3,062 05/08/2012, 06:59 PM Last Post: sheldonison help with a distributivity law JmsNxn 3 5,795 09/22/2011, 03:32 PM Last Post: JmsNxn Base 'Enigma' iterative exponential, tetrational and pentational Cherrina_Pixie 4 12,916 07/02/2011, 07:13 AM Last Post: bo198214 Two exponential integrals Augustrush 2 6,507 11/10/2010, 06:44 PM Last Post: Augustrush HELP NEEDED: Exponential Factorial and Tetrations rsgerard 5 11,694 11/13/2009, 02:27 AM Last Post: rsgerard Exponential factorial mike3 3 8,325 10/07/2009, 02:04 AM Last Post: andydude Additional super exponential condition bo198214 4 7,886 10/21/2008, 03:40 PM Last Post: martin exponential polynomial interpolation Gottfried 3 9,538 07/16/2008, 10:32 PM Last Post: andydude

Users browsing this thread: 1 Guest(s)