08/31/2007, 07:04 PM
Wow, I'm not so impressed at the formulas as I am finding that they converge. I played around with dozens of infinite systems before I found that my matrix equation for the super-logarithm did converge. I would like to point out, as Jay seemed to point out, that the first formula you gave was obvious. The second is not so obvious (and I still have my doubts until I can derive it), but what is obvious to me, is that given the first two formulas, the others are easily proven.
But until I can derive the second formula, I'll assume that it's magic
On a slightly different note, your TetraExpPrime reminds me of Szekeres' mention of the Julia functional equation (FE), and I think its actually \( 1/f'(x) \) and not f'(x), but thats not the point. The point is that the first derivative of tetration or the super-logarithm is much more fundamental than the function itself, because it by-passes the requirement \( {}^{0}x = 1 \), and thus would work for any shifted-tetration as well.
Andrew Robbins
Szekeres' overview of functional equations
hyper4geek Wrote:TetraExp[x] == TetraExpPrime[x] / TetraExpPrime[x-1]
TetraExp[x] == ProductLog[TetraExpPrime[x+1] / TetraExpPrime[x-1]]
But until I can derive the second formula, I'll assume that it's magic
On a slightly different note, your TetraExpPrime reminds me of Szekeres' mention of the Julia functional equation (FE), and I think its actually \( 1/f'(x) \) and not f'(x), but thats not the point. The point is that the first derivative of tetration or the super-logarithm is much more fundamental than the function itself, because it by-passes the requirement \( {}^{0}x = 1 \), and thus would work for any shifted-tetration as well.
Andrew Robbins
Szekeres' overview of functional equations
