x↑↑x = -1

[sheldonison]

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Since For x~=2.6918099719192 + 0.62660048483655i, sexp_x(x)~=-1 apparantly , I wonder about 2 things

-- apart from the fact that I need to learn more about complex base iterations ! --

1) Is this x~=2.6918099719192 + 0.62660048483655i the only solution ??

2) What happens with x~=2.6918099719192 - 0.62660048483655i ??
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And as suggested by question nr 2 :

x^^x = conj( conj(x)^^conj(x) ) or similar ??

The conjugate of x would also be a solution, with x^^x = -1. The f=conj(f(conj(x))) equation is correct since the function is real valued at the real axis. The solution at hand for sexp_x(conj(x))~=3000+5500i Though that probably wasn't your question...

I would expect an infinite number of solutions. Seeing as individually each tetration base solution takes on all values, then probably x^^x=y always has a solution, but finding them isn't easy.

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But the main question is : is x^^x analytically continuable to the entire complex plane or does it have a natural boundary ?

We might want to move this to the tetcomplex.gp thread. This complex base tetration stuff was originally inspired by Mike3 solving one particular complex base. I took it a little farther, to try and explore the branch point at exp(1/e), and understand how tetration would work for real bases<exp(1/e), which also turns out to be very interesting; the solution extended from real tetration is not real valued there. So, there's this boundary, called the Shell Thron boundary, where the two primary tetration fixed points switch from two repelling fixed points, to one repelling and one neutral. Its hard to calculate tetration on the boundary, since one of the fixed points is neutral, and has no superfunction, but it turns out the bipolar complex tetration solution is still analytic there, except of course for at eta=exp(1/e). But, if you circle around eta almost one full iteration, you will eventually cross the ShellThron a second time, and that appears to be the natural boundary for complex tetration, so I would expect that boundary to be the boundary for x^^x as well. Starting with a real base>eta, you can go almost one full revolution in either direction. Adrien Douady conjectured a similar boundary for iterations of: \( x^2+\frac{1}{4}+\epsilon \;\; \) See post#17, http://math.eretrandre.org/tetrationforu...416&page=2 Beyond that, there are other known singularities at b=1 and b=0. I got as far as conjecturing a solution for e^^-e, but I have never solved a a base with real(b) negative.

But thats not all , maybe I need to learn more about complex base iterations , but I know there are multiple solutions right ?

So what answers do the other methods give ?

Many questions ...

I understand the kneser method for real bases > eta , but not for complex.

for real bases we map the reals to the real line , but what do we do for the complex bases ??
Or is that an irrelevant question ?
Sorry complex bases still confuse me.

As for as we can tell, Kouznetsov's method agrees with Kneser. The Kneser numeric method is a lot easier to extend to arbitrarily high precision levels. For complex tetration, we are interesting in analytically continuing Kneser so there is theoretically only one solution. The complex base solution is pretty weak, theoretically, though Henryk says you can rigorously show it should exist using the measurable Riemann mapping Theorem. Theoretically, research in complex dynamics might be applicable here too.

Basically, you still have two fixed points. I use a pair of theta mapping for the two primary fixed points. The solution you're looking for is
1) analytically continuable from real base tetration, as you slowly modify
the base, avoiding the singularity at eta.
2) sexp(-1)=0, sexp(0)=1, sexp(1)=b
3) converges to the superfunction for one fixed point in the upper half of the complex plane (theta mapping number 1)
4) converges to the superfunction for the other fixed point in the lower half of the complex plane (theta mapping number 2)

The programming for tetcomplex.gp is not nearly as rock stable as the real base kneser.gp program. There are lots of numerical problems at or near the ShellThron boundary, for example, and with using the wrong logarithmic branch, and at the singularity at eta, and with the algorithm sometimes not converging as precision is increased.