Infinite tetration of $(-1)$?

Question

For all positive integers $n$, $^n(-1) = -1$, thus I thought ${^{\infty}}(-1)$ could be $\displaystyle \lim_{n \to \infty} {^n}(-1) = -1$.

But $\dfrac{W(-\ln z)}{-\ln z}$, analytic continuation of infinite power tower, gives some imaginary number.

Which one is correct for this tetration?

Answer 1

Lambert-W gives the real value $-1$ when we use the branch at $k=1$ (in some software at $k=-1$).

Examples:

1. W|A gives for exp(-productlog(1,-log(-1))) the result: -1
2. An implementation in Pari/GP gives for exp(-LambertW(-log(-1),-1)) the result -1.000000000000000 + 3.300 E-213*I
   (the spurious imaginary part is due to internal float precision calculations and no default rounding/truncation is performed)

Your value is given when the default branch of the Lambert-W is chosen.

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Using a Pari/GP implementation of the Lambert-W I find the following exemplaric list (from an infinite number of solutions) :

$$\small \begin{array} {rrl} k & \exp(-\operatorname{LambertW}(-\log(-1),k)) \\ \hline -9 & 17.98372743-0.9201621357*I \\ -8 & 15.98243823-0.8826775326*I \\ -7 & 13.98089425-0.8401770770*I \\ -6 & 11.97900898-0.7911056923*I \\ -5 & 9.976653691-0.7330485254*I \\ -4 & 7.973635250-0.6619491593*I \\ -3 & 5.969690012-0.5701679520*I \\ -2 & 3.964786505-0.4404083501*I \\ -1 & 1.964999038-0.2169439255*I \\ 0 & 0.2660365993+0.2942900219*I & *** \\ 1 & -1.000000000+3.308E-213*I \\ 2 & -2.962792822-0.3479062027*I \\ 3 & -4.967312786-0.5118935159*I \\ 4 & -6.971795660-0.6193684989*I \\ 5 & -8.975242573-0.6994852445*I \\ 6 & -10.97790036-0.7634014098*I \\ 7 & -12.98000110-0.8165873435*I \\ 8 & -14.98170272-0.8621368113*I \\ 9 & -16.98311045-0.9019717088*I \end{array} $$ The marked entry (with positive or negative imaginary component) should be the one that you've got yourself, correct?

Answer 2

If we think of the infinite power tower as a limit over a real height, I think the -1 case is comparable to the limit of $\cos 2 \pi n$. For integer $n$, it is always 1, but the limit over real $n$ doesn't exist.

If you define a crude version of fractional tetrarion -- eg $^{2.5}(x)=x^{x^\sqrt x}$, then for $0<x\le e^{1/e}$, the limit of tetrarion over the reals equals the one over the integers.

On the other hand, this breaks down for $-1$. To summarize, I think the limit existing is tied to some definition of fractional tetrarion. Even though I don't think that's a well established thing, it's hard to imagine it being "nice" for negative numbers.