11/19/2007, 10:52 AM
Hej Andy,
Thanks, the second I can access, the first is pay for article- i can not afford to pay for articles as I sometimes need too many, and then some of those are not the right ones... I will have a look.
That is interesting, that is one of the reasons I thought the value could be 1/e -or e-e or (1/e)^(1/e) as h(1/e) = Omega -> I hoped there is a way to attribute value (or many different values) even if h(0) diverges as tetration. So that value would not be tetration, but - a value.In a way similar to series +1-1+1-1 ........ which are not summable in ordinary sense and than use these non-rigorous values of h(0) , h (z>e^1/e) to find new relations.
For that, may be there is a need/possibility for a starter to find an analoque of geometric series sums or values of a type 1/(1-x) or 1/(1+x) in tetration,which would work in converge nce region but give values outside it. I think Gotfrieds formula goes in that direction when all branches will be found. Just an idea.
Ivars
andydude Wrote:To answer your specific question, you should probably read either Knoebel's Exponentials Reiterated or McDonnell's
Some Critical Points of the Hyperpower Function since they both talk about this.
Thanks, the second I can access, the first is pay for article- i can not afford to pay for articles as I sometimes need too many, and then some of those are not the right ones... I will have a look.
Quote:Near zero, \( h(x) = {}^{\infty}x \) can be defined in one of three ways: as the limit of odd towers as the height tends to infinity (giving \( h_O(0) = 0 \)), as the limit of even towers as the height tends to infinity (giving \( h_E(0) = 1 \)), or as the inverse function of \( f(x) = x^{1/x} \) (giving \( h_I(0) = 0 \)). Each one of these definitions gives a different answer as x approaches 0. The strange thing is that all definitions are equivalent for \( e^{-e} < x e^{1/e} \).
Andrew Robbins
That is interesting, that is one of the reasons I thought the value could be 1/e -or e-e or (1/e)^(1/e) as h(1/e) = Omega -> I hoped there is a way to attribute value (or many different values) even if h(0) diverges as tetration. So that value would not be tetration, but - a value.In a way similar to series +1-1+1-1 ........ which are not summable in ordinary sense and than use these non-rigorous values of h(0) , h (z>e^1/e) to find new relations.
For that, may be there is a need/possibility for a starter to find an analoque of geometric series sums or values of a type 1/(1-x) or 1/(1+x) in tetration,which would work in converge nce region but give values outside it. I think Gotfrieds formula goes in that direction when all branches will be found. Just an idea.
Ivars