07/01/2009, 02:27 AM
(This post was last modified: 07/01/2009, 02:40 AM by Arthur Rubin.)
I thought I'd introduce myself, and explain how I got here.
I found this forum through Wikipedia, my interest in tetration dates back a long time, although I didn't know the name.
Sometime when I was a graduate student in mathematics at CalTech, Richard Feynman used to wander into the math lounge and ask weird questions. Most of them either had obvious answers or obviously (to me, anyway), had no answers. However, one question that he asked was whether there is a "natural" increasing functions f and g such that
(a) f(f(x)) = e^x,
(b) g(x+1)=e^g(x)
(Again, it was obvious even then to me, that if there is a solution of (b) for g, then
: f(x) = g(1/2+g^(-1)(x))
is a solution of (a) , and that there are many C^(infinity) solutions of (b)with g(0)=1, and g defined on (-2, +infinity). (I might have selected g(0)=0, but it doesn't really matter, right.)
In the notation of this forum, g(x) = e^^x. As I pointed out over on Wikipedia, if there is a real-analytic solution of (b), there are many, as if h is real-analytic, period 1, and "small" (to satisfy "increasing"), then g(x+h(x)) also satisfies condition (b).
And I don't really know much more than that. I hope that, together with some of the other posters, we can find some sort of "natural" tetration.
I found this forum through Wikipedia, my interest in tetration dates back a long time, although I didn't know the name.
Sometime when I was a graduate student in mathematics at CalTech, Richard Feynman used to wander into the math lounge and ask weird questions. Most of them either had obvious answers or obviously (to me, anyway), had no answers. However, one question that he asked was whether there is a "natural" increasing functions f and g such that
(a) f(f(x)) = e^x,
(b) g(x+1)=e^g(x)
(Again, it was obvious even then to me, that if there is a solution of (b) for g, then
: f(x) = g(1/2+g^(-1)(x))
is a solution of (a) , and that there are many C^(infinity) solutions of (b)with g(0)=1, and g defined on (-2, +infinity). (I might have selected g(0)=0, but it doesn't really matter, right.)
In the notation of this forum, g(x) = e^^x. As I pointed out over on Wikipedia, if there is a real-analytic solution of (b), there are many, as if h is real-analytic, period 1, and "small" (to satisfy "increasing"), then g(x+h(x)) also satisfies condition (b).
And I don't really know much more than that. I hope that, together with some of the other posters, we can find some sort of "natural" tetration.
