Should tetration be a multivalued function?

[marraco]

Let r be a real number 0<r<1

We can numerically calculate c₍ᵣ₎=ʳa and c₍₋ᵣ₎=⁻ʳa with Sheldonison's knesser-gp.

No proof, but numerically:
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{(^{r+1}a)^{^{-(r+1)}a}=(^{-r}a)^{^{r}a}} \)

and in general


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{(^{r+(n+1)}a)^{^{-(r+(n+1))}a}=(^{-(r+n)}a)^{^{r+n}a}} \)

This remains valid for -(r+n)<-2, where the iterated logarithms are complex numbers.


This gives a natural way to define °a for the Z branch. °a should be the value that makes:

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{^{-1}a\,^{^1a}\,=\,^1a\,\,^{^{-1}a}\,=\,^{0}a\,^{^0a}\, \; or \; (log_a{^0a})\,^{(a^{(^0a))}=(a^{(^0a)})\,^{(log_a{^0a})} \)

then it also satisfies:

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{(^{r+(n+1)}a)^{^{-(r+(n+1))}a}=(^{-(r+n)}a)^{^{r+n}a}} \)


Example (for the main branch)



For the Z branch