• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Should tetration be a multivalued function? marraco Fellow Posts: 93 Threads: 11 Joined: Apr 2011 01/09/2016, 05:14 PM (This post was last modified: 01/11/2016, 06:25 PM by marraco.) (01/09/2016, 06:20 AM)sheldonison Wrote: (01/08/2016, 06:26 PM)marraco Wrote: .... No proof, but numerically: $\vspace{15}{(^{r+1}a)^{^{-(r+1)}a}=(^{-r}a)^{^{r}a}}$ ok, first lets define y and z as follows: $y=\; ^{-(r+1)}a \;\;\;$ $z=\; ^{r}a \;\;\;$ Then substitute these values of y and z into the Op's equation above, noting that $^{(r+1)}a = a^z\;\;\;\; ^{-r}a = a^y$ $(a^z)^{y}=(a^y)^z\;\;\;$ This is the Op's equations with the substitutions $(a^z)^{y}=(a^y)^z=a^{(y\cdot z)}\;\;\;$ this equation holds for all values of a,y,z^^ Good. That's far more elegant that my reasoning, which used the commutative x^ln(y) and goofy induction. Maybe is the key to calculate the derivative at the origin? (taking limit of r → 0) Here is a graphic illustrating the equality. The exponentiation of the blue arrow is equal to the one in the red arrow. The expression inside the rectangles are equal. By induction, if we invert the red arrow, it is equal to a larger arrow; I mean $\vspace{20}{(^{r+(n+1)}a)^{^{-(r+(n+1))}a}=(^{-(r+n)}a)^{^{r+n}a}}$. I wrote it thinking that n was entire, but since r is real, and °a is arbitrary, it is obvious that is for any real n. But is even more general, because this is valid on all branches, and for any definition of °a. Graphically, you can move the arrows to the left or right, and the equality remains valid. For example, $\vspace{20}{^2a\,^{^3a}\,=\,^4a\,^{^1a}}$ and also $\vspace{20}{^1a\,^{^4a}\,=\,^5a\,^{^0a}}$, $\vspace{20}{^0a\,^{^5a}\,=\,^6a\,^{^{-1}a}}$, etc. I have the result, but I do not yet know how to get it. « Next Oldest | Next Newest »

 Messages In This Thread Should tetration be a multivalued function? - by marraco - 01/03/2016, 11:24 PM RE: Should tetration be a multivalued function? - by marraco - 01/03/2016, 11:32 PM RE: Should tetration be a multivalued function? - by andydude - 01/03/2016, 11:55 PM RE: Should tetration be a multivalued function? - by andydude - 01/04/2016, 12:01 AM RE: Should tetration be a multivalued function? - by marraco - 01/04/2016, 12:08 AM RE: Should tetration be a multivalued function? - by andydude - 01/04/2016, 12:35 AM RE: Should tetration be a multivalued function? - by marraco - 01/04/2016, 03:57 AM RE: Should tetration be a multivalued function? - by marraco - 01/08/2016, 06:26 PM RE: Should tetration be a multivalued function? - by marraco - 01/08/2016, 11:05 PM RE: Should tetration be a multivalued function? - by sheldonison - 01/09/2016, 06:20 AM RE: Should tetration be a multivalued function? - by marraco - 01/09/2016, 05:56 PM RE: Should tetration be a multivalued function? - by marraco - 01/14/2016, 04:24 AM RE: Should tetration be a multivalued function? - by tommy1729 - 01/10/2016, 03:37 AM RE: Should tetration be a multivalued function? - by marraco - 01/11/2016, 09:28 PM RE: Should tetration be a multivalued function? - by andydude - 01/13/2016, 08:24 AM RE: Should tetration be a multivalued function? - by sheldonison - 01/13/2016, 01:37 PM

 Possibly Related Threads... Thread Author Replies Views Last Post New mathematical object - hyperanalytic function arybnikov 4 1,061 01/02/2020, 01:38 AM Last Post: arybnikov Is there a function space for tetration? Chenjesu 0 673 06/23/2019, 08:24 PM Last Post: Chenjesu Degamma function Xorter 0 1,125 10/22/2018, 11:29 AM Last Post: Xorter Introducing new special function : Lambert_t(z,r) tommy1729 2 4,013 01/10/2016, 06:14 PM Last Post: tommy1729 Tommy-Mandelbrot function tommy1729 0 2,182 04/21/2015, 01:02 PM Last Post: tommy1729 The inverse gamma function. tommy1729 3 8,207 05/13/2014, 02:18 PM Last Post: JmsNxn entire function close to sexp ?? tommy1729 8 10,367 04/30/2014, 03:49 PM Last Post: JmsNxn Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... Gottfried 5 7,543 07/17/2013, 09:46 AM Last Post: Gottfried Proof Ackermann function extended to reals cannot be commutative/associative JmsNxn 1 3,299 06/15/2013, 08:02 PM Last Post: MphLee Hyperzeta function Balarka Sen 7 8,755 05/08/2013, 01:36 PM Last Post: Balarka Sen

Users browsing this thread: 1 Guest(s)