08/12/2007, 10:52 PM
[quote=jaydfox]
In Andrew Robbins paper [1], he mentions three properties, two of which are applicable to tetration:
Property 1. Iterated exponential property
\( ^{y}x=x^{\left({ ^{y-1} x }\right)} \) for all real y.
Property 3. Infinite differentiability property
\( \large f(x)\ \text{is}\ C^\infty\ \equiv\ D^k f(x) \) exists for all integer k.
At the bottom of page three, Andrew goes on to say:
[QUOTE]It is the goal of this paper, however, to show that these properties are sufficient to find such an extension, and that the extension found will be unique.[/QUOTE]
It doesn't seem to be, tho.
To complicate matters, I fixed the piecewise differentiability problem with the frac extension and it turns out that the original construction can be made \( C^\infty \), when the base is e.
I also posted a note on sci.math.research
Details here
In Andrew Robbins paper [1], he mentions three properties, two of which are applicable to tetration:
Property 1. Iterated exponential property
\( ^{y}x=x^{\left({ ^{y-1} x }\right)} \) for all real y.
Property 3. Infinite differentiability property
\( \large f(x)\ \text{is}\ C^\infty\ \equiv\ D^k f(x) \) exists for all integer k.
At the bottom of page three, Andrew goes on to say:
[QUOTE]It is the goal of this paper, however, to show that these properties are sufficient to find such an extension, and that the extension found will be unique.[/QUOTE]
It doesn't seem to be, tho.
To complicate matters, I fixed the piecewise differentiability problem with the frac extension and it turns out that the original construction can be made \( C^\infty \), when the base is e.
I also posted a note on sci.math.research
Details here