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Possible continuous extension of tetration to the reals - Printable Version

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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:

and



Where and

This leads to a continuous function that looks like a very steep exponential graph.

It also leads to several identities:






Please tell me your thoughts on this, thanks.

~Dasedes