andydude Wrote:Now that that's proven, anything thats true about one function should be true about the other function, since the relationship is linear. However, Trappmann said that \( e^x - 1 \) has a continuous iterate that fails to converge for non-integers. This could be disastrous for tetration (another reason I didn't want to post this). But it also means that since the function \( y = b^{1/b} \) is unique for b=e, it means that base-\( e^{1/e} \) tetration is uniquely defined, even if its series doesn't converge.
Andrew Robbins
Hey, can you point me to where Trappmann said that \( e^x - 1 \) fails to converge for non-integers? (Or did you mean non-integer x?) In my own limited experimentation, it looks like it should converge quite nicely, at least on a small enough radius and limiting ourselves to the reals (haven't tried complex numbers yet). Integer iteration counts should extend the function out to the rest of the reals.
But I'm only looking at the first 15 or so terms of the series. It seems pretty well behaved and very well-defined, but maybe I'm missing something?
Edit: Never mind, I found it here:
http://math.eretrandre.org/tetrationforu...d=28#pid28
I'll take a look at his reference, if I can get a hold of it.
