Possible continuous extension of tetration to the reals - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Possible continuous extension of tetration to the reals (/showthread.php?tid=1099) Possible continuous extension of tetration to the reals - Dasedes - 10/10/2016 I've been sort of obsessed with Tetratipn since I first heard about it, and I've scoured the internet trying to learn all that I could about it. I came across this oeis Talk: Iterated Logarithms recently and the idea of a weighted function to deal with non-Natural numbers struck me as interesting. I've played with his idea and came up with this: $^ya = {a_{_0}}^.^{.}^.^{(a_{h-1} - ((a - 1) - r))}$ and $log_b*(^ya) = \lceil slog_b(^ya) \rceil = \lceil y \rceil$ $slog_b(^ya) = log_b*(^ya) - r$ Where $r = y - \lfloor y \rfloor$ and $h = \lfloor y + 1 \rfloor$ This leads to a continuous function that looks like a very steep exponential graph. It also leads to several identities: $slog_b(^ya) = yslog_b(a)$ $^0a = a_0 - (a - 1) = a - a + 1 = 1;$$h = \lfloor 0 + 1 \rfloor = 1$ $^{-y}a = \frac {1}{a_{_0}^.^{.}^.^{(a_{h-1} - ((a - 1) - r))}}$ Please tell me your thoughts on this, thanks. ~Dasedes