Hey jaydfox !... this would be a breakthrough WOW!! - generations of mathematicians have sought for it - pity it is so complicated, i wonder if inventing a new function would make it easier ?... Lambert invented one, too, to make certain equations easier to solve *G*
Maybe you will become famous :-)
I continued my research on a newly formed forum dedicated to tetration and related research.
Alas, after further probing, I discovered that my solution wasn't all that great after all.
For those interested, extending tetration to the reals requires a couple things at the minumum:
1. Iterated exponential property
If we have a tetration function T_b(x) for a base b, then T_b(x) = b^T_b(x-1) for all real x. This is a property of a more general class of superexponentiation relations, of which tetration is a specific instance such that T_b(0)=1. It follows that T_b(-1)=0 and T_b(-2) = negative infinity
2. Infinite differentiability property
The function T_b is continuous for real x > -2, and it can be differentiated infinitely many times (with the derivatives being continuous over the same domain)
My solution satisfied these two conditions. In my haste, I found a simple and in some ways beautiful way to constuct a solution that satisfied the two conditions. However, there are infinitely many such solutions, and mine isn't all that special aside from its relative simplicity. (And yes, that might seem rather tongue-in-cheek, because it may look complicated, but it's not really that complicated once you understand how it works.)
As it turns out, we need at least a third condition to uniquely define "the" solution. I have a couple really good such conditions, and luckily both seem to favor the same solution. And no, it's not my solution. It appears that "the" solution is the inverse of the matrix solution of the Abel functional equation. For more information on the Abel function, see this post by Andrew Robbins (only hardcore math geeks need follow the link :chasses: ):
http://math.eretrand...read.php?tid=27Andrew Robbins independently derived a matrix-based solution:
http://tetration.itgo.com/paper.htmlAfter spending weeks studying his solution, I figured out that it's just an alternate method of calculating the basic Abel matrix solution. But seeing it solved both ways really convinced me that it's valid, and knowing that it satisfies both of my additional criteria for uniqueness just caps it off.
For the curious, my two additional criteria (which may or may not yield the same solution):
1. Odd derivatives should be non-negative (perhaps even strictly positive?) for real x > -2. This ensures that all odd derivatives are convex. Even the most insignificant deviation from "the" solution would result in negative values in some of the odd derivatives.
1b. Odd derivatives should be log-convex for real x > -2. I haven't studied this condition enough to know if it's stricter than just being convex. For a single derivative, log-convexity is stricter, but I think that if all odd derivatives are convex, then they are all log-convex. I need to prove or disprove this at some point.
2. If we extend the real solution to complex numbers (using the power series and analytic continuation), then there should be logarithmic singularities at the primary fixed points of the underlying exponential funciton. Even the most insignificant deviation from "the" solution would result in the singularities becoming non-logarithmic.