Should tetration be a multivalued function?

[sheldonison]

...The previous functions discussed earilier are 3-initialized and 5-initialized, which makes them, not tetration, by definition. However, if there is an analytic continuation of the 1-initialized superfunction that overlaps with the 3-initialized superfunction, AND if on the overlap f(0) = 3, then they can be considered branches of the same function. But until that is proven, I don't think it's accurate to say that they're all "tetration". They are, however, iterated exponentials in the sense that they extend \( \exp_b^n(3) \) to non-integer n. And so I would probably write these functions as \( \exp_b^n(1),\, \exp_b^n(3),\, \exp_b^n(5) \) instead of saying that \( {}^{n}b \) is a multivalued function that returns all three.

ok, I was thinking of posting something similar, but I wasn't sure if it was off topic or not. From my perspective, the definition of tetration begins with real valued b>eta, where the base b \( b>\eta \;\;\; b>\exp(\frac{1}{e}) \). These bases require a Kneser mapping, to get a unique real valued function at the real axis, so that:

\( \text{tet}_b(z) =\;^z b= \exp_b^z(1)\;\;\; \text{tet}_b(0)=1, \; \text{tet}_b(1)=b;\;\;\; \) and tet(z) grows superexponentially as z gets larger

The Op is working with iterated exponentials using b=sqrt(2). These iterated functions are valid super-exponentials. But it turns out they are not Kneser tetration, in the above sense. Maybe that's not an important distinction. Anyway, it turns out, we can extend tetration to complex bases, and experiment with how the function behaves as we vary the base around \( \eta \). I didn't know any of this when I wrote kneser.gp for real valued tetration. But I have since written two different pari-gp programs posted on this site to experiment with complex base tetration. One is called tetcomplex.gp, and the other more recent effort calculates the Abel/slog function for arbitrary complex bases, fatou.gp The two pari-gp programs agree with each other. And they both agree that when you rotate 180 degrees around eta, the function you get is no longer real valued at the real axis! So there is such a thing as a well defined tetration \( b=\sqrt{2}\; \text{tet}_b(z)\;\; \) that is an analytic continuation of Kneser type tetration. But guess what. Its not real valued at the real axis. Its some wierd function, that is almost real valued. In the upper half of the complex plane, the function gets arbitrarily close to behaving like the attracting fixed point of 2 function whose period is ~=17.1i , and in the lower half of the complex plane, it gets arbitrarily close to behaving like the repelling fixed point function, whose period is ~=19.2i. At the real axis, \( \text{tet}_{\sqrt{2}}(0.5) = \)1.24362162766852 - 1.18899611608401 E-48*I

It is much easier to see the effect with a base like b=1.25, where the default settings and precision are sufficient for the calculations.

Code:

\r \fatou.gp
sexpinit(1.25);
sexp(0.5)
1.16359536196501 - 2.01428234045867 E-16*I