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+--- 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