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 Should tetration be a multivalued function? marraco Fellow Posts: 93 Threads: 11 Joined: Apr 2011 01/03/2016, 11:24 PM (This post was last modified: 01/04/2016, 12:08 AM by marraco.) When we have a tetration $\vspace{15}{y_{x}=\,^xa}$ with his base "a" between $\vspace{25}{1< a \leq e^{\frac{1}{e}}}$, it tends to an asymptote value $\vspace{15}{y_{\infty}}$ such that $\vspace{25}{a^{y_{\infty}}={y_{\infty}}}$. But that depends on the definition $\vspace{15}{^0a=1}$. If it were defined $\vspace{15}{^0a=y_{\infty}}$, we would get an horizontal line (drawn with red dashes in the graphic down). The interesting part is when we define $\vspace{15}{^0a}$ a bit larger than $\vspace{15}{y_{\infty}}$, the tetration also converges to the same asymptote to the right, and another asymptote to the left, at the limit $\vspace{15}{y_{-\infty}}$, defined by $\vspace{25}{a^{y_{-\infty}}={y_{-\infty}}}$. We get a Z-shaped tetration function (drawn in green), contained between $\vspace{15}{y_{-\infty}}$ and $\vspace{15}{y_{\infty}}$, which I call the Z curve. The problem is that the Z curve is not uniquely defined. It depends on the value choosen for $\vspace{15}{^0a}$. Any value between $\vspace{20}{{y_{\infty}} < ^0a < {y_{-\infty}}}$ is valid, and the only difference this choice make, is the horizontal displacement on the curve. I drew the Z curve matching the origin with his inflection point (roughly). The upper asymptote is also a non trivial function, and there is also another non trivial possible upper branch, obtained by choosing °a>-oo (drawn in blue line). The question is, what would be convenient values for the definition of $\vspace{15}{^0a}$ for the green and blue branches? If we take $\vspace{15}{a=\sqrt{2}}$, $\vspace{15}{^0a}$ may be multivalued at x=0: $\vspace{15}{^0(\sqrt{2})=(1,2,\; 2. The inflection point in the Z curve is near to 3. Maybe that base has all integer values for °a? (°1,41421356 = (1,2,3,4,5) ) Note how the blue branch resembles the function $\vspace{15}{e^x}$. 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

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