Intereting Posts

A commutative ring $A$ is a field iff $A$ is a PID
What does it mean to say a boundary is $C^k$?
Given that a,b,c are distinct positive real numbers, prove that (a + b +c)( 1/a + 1/b + 1/c)>9
Finding the Fourier series of a piecewise function
Proving that $|CA|+|CB|=2|AB|$ in a general $ABC$ triangle
Prove: The positive integers cannot be partitioned into arithmetic sequences (using Complex Analysis)
Prove a number is composite
Function $\mathbb{R}\to\mathbb{R}$ that is continuous and bounded, but not uniformly continuous
Are Continuous Functions Always Differentiable?
is uniform convergent sequence leads to bounded function?
Proving ${p-1 \choose k}\equiv (-1)^{k}\pmod{p}: p \in \mathbb{P}$
Integer solutions for $x^3+2=y^2$?
Continuous bounded functions in $L^1$
Hom of finitely generated modules over a noetherian ring
How to prove that this solution of heat equation is not a tempered distribution?

Let $z \in \mathbb{C}$ and let $W$ be the Lambert $W$ function. In this post it is shown that if $|W(-\ln z)| > 1$ then the infinite power tower $z^{z^{z^{z^…}}}$ does not converge, that is $|W(-\ln z)| \leq 1$ is a necessary condition for the convergence of $z^{z^{z^{z^…}}}$.

Here I would like to show that $|W(-\ln z)| < 1$ is a sufficient condition, that is if $|W(-\ln z)| < 1$ then $z^{z^{z^{z^…}}}$ is convergent.

- Determine the number of zeros of the polynomial $f(z)=z^{3}-2z-3$ in the region $A= \{ z : \Re(z) > 0, |\Im(z)| < \Re(z) \}$
- How to prove $f(z) = \sum_{n=-\infty}^{\infty}e^{2\pi inz}e^{-\pi n^2}$ has a unique zero inside a unit square in the 1st quadrant.
- Intersections of the level curves of two (conjugate) harmonic functions
- Maximum of $\frac{\sin z}{z}$ in the closed unit disc.
- Using Residue theorem to evaluate $ \int_0^\pi \sin^{2n}\theta\, d\theta $
- For a fixed complex number $z$, if $z_{n}=\left( 1+\frac{z}{n}\right)^{n}$. Find $\lim_{n \to \infty}|z_{n}|$

- Is the square root of a negative number defined?
- Show how to calculate the Riemann zeta function for the first non-trivial zero
- Value of Summation of $\log(n)$
- Applications of complex numbers to solve non-complex problems
- Integrating $\int_0^\infty \frac{\ln x}{x^2+4}\,dx$ with residue theorem.
- The continuity assumption in Schwarz's reflection principle
- Meromorphic on unit disc with absolute value 1 on the circle is a rational function.
- For a fixed complex number $z$, if $z_{n}=\left( 1+\frac{z}{n}\right)^{n}$. Find $\lim_{n \to \infty}|z_{n}|$
- There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x_{1}+x_{2}+\cdots+x_{k}|\ge 1$
- Does $i^4$ equal $1?$

The key concept is here the “Shell-Thron-region”. In articles in the previous century initially W. Thron and later D. Shell based on Thron’s work proved that if you have a complex base, say $b$ such that $b=t^{1/t}$ or, with $u=\log(t)$, such that $b=\exp(u \exp(-u))$ then the infinite power tower converges if $|u| \lt 1$ and the point of convergence is $t$. (See my earlier picture in MSE where I’ve related those 3 variables with each other)

The numerical values given in Yiannis Galidakis’ answer have $|u|=1-\varepsilon$ so the iteration should converge although very slowly. I found, that a nice picture occurs if you separate the trajectory in *4*, or even better: *72* subtrajectories. With Pari/GP and *800* digits precision you get a nice shape which has some “fractal-like” or “snowflake-like” border. I’ve done the iterations from $z_0=1$ to up to *80 x 72* iterations so each partial curve has *80* points, nearly neighboured with each other – and each pair of neighboured points of the same color has distance of *72* iterations; for a real good image one should proceed to at least *72^3 x 72* points to get a valid impression that this strangely shaped curve really contracts. See a q&d picture made with Excel using values made with Pari/GP, *800* digits precision:

One recognizes *4* segments which together do about one round. These are the first four segments of *72* segments, (where the *5* ‘th would nearly overlap the first, the *6* ‘th nearly the second and so on, but are not shown here to keep the picture clean). The brown segment is the *32* ‘th and its small additional excess from the blue one shows that the expected contraction is -at least- not smooth.

I’ve no nerve to increase the number of points at the moment (it’s night here), possibly my hints give enough ideas to proceed on your own.

** [update]** I couldn’t stop to try to discern the convergence. It appears, that not only in steps of 72 the iterates are tight together, but that it needs 322 of such 72-steps to fill one round of the curve. So I took an arbitrary initial value from my existing list, $\small y_0=-0.5602531521 – 0.6868631844 I$, then iterated 322*72=23184 times to arrive at $ \small y_{23184} \approx -0.5602563718 – 0.6868510240 I$ and proceeded

```
real y_k imag y_k | y_k - t| =distance to fixpoint
-0.5602563718 -0.6868510240 0.8863698615
-0.5602654611 -0.6868245642 0.8863551032
-0.5602477332 -0.6867936307 0.8863199274
-0.5602391855 -0.6867763400 0.8863011262
-0.5602265922 -0.6867593629 0.8862800106
-0.5602215274 -0.6867495245 0.8862691855
-0.5602178010 -0.6867358203 0.8862562109
-0.5602148750 -0.6867278280 0.8862481684
-0.5602175553 -0.6867173579 0.8862417497
-0.5602183774 -0.6867059453 0.8862334262
-0.5602266751 -0.6866972799 0.8862319570
-0.5602460790 -0.6866801357 0.8862309393
-0.5602492634 -0.6866452713 0.8862059387
-0.5602499101 -0.6866233340 0.8861893503
-0.5602465541 -0.6865997931 0.8861689891
-0.5602452598 -0.6865858545 0.8861573713
-0.5602473046 -0.6865693793 0.8861458993
-0.5602478202 -0.6865574553 0.8861369869
-0.5602548561 -0.6865457844 0.8861323930
-0.5602616629 -0.6865325382 0.8861264340
```

*[end update]*

That is different with values *b* where the according value of *u* is $|u|=1$ and thus lie on the boundary of the complex unit disk. An example has been given in the comment at Yiannis Galidakis’ answer with *u* as some complex unit root. Then we have no convergence and the curve (with roughly same shape as the beginning of the shown curve) does *not* contract but has its trajectory “stationary” – I called this, when I’ve seen it first time, “equator” because it reminds me to the meridians on a globus – not disappearing away from and also not contracting towards the fixpoint, but of course there are established technical terms for such -thanks to Yiannis pointing me to this in a comment last night.

P.s.: to improve numerical stability and computing speed when many iterations are needed, use the following conjugacy relation:

the original iteration demands

- use some $\small z_0$,

compute $\small z_{k+1} = b^{z_k}$ and iterate

until some $\small z_n$ .

You can do the following replacement:

- use the same $\small z_0$,

compute $\small y_0=(z_0/t)-1$ ,

compute $\small y_{k+1}=\exp(u \cdot y_k) – 1$ for as many iterations as before

to get $\small y_n$,

then compute $\small z_n = (y_n+1) \cdot t$

$ \qquad \qquad $ with $\small t=\exp(-W(-\ln(b)))$ and $\small u=\ln(t)=-W(-\ln(b))$

Here I would like to show that $|W(−\ln(z))|\le 1$ is also a sufficient condition, that is if $|W(−\ln(z))|\le 1$ then $z^{z^{z^{\ldots}}}$ is convergent.

It’s not true. Take $c=2.043759690+0.9345225945i$. Then (with some Maple code:)

```
restart;
with(plots);
F := proc (z, n)#power tower recursively defined
option remember;
if n = 1 then z else z^F(z, n-1)
end if
end proc;
W := LambertW;
c := 2.043759690+.9345225945*I;
evalf(abs(W(-ln(c))));
```

0.99999999

```
L := [seq(evalf(F(c, n)), n = 1 .. 100)];
complexplot(L, style = point);
```

Here’s the list of values $c,c^c,c^{c^c},\ldots$ plotted against the Complex plane:

- What is the average of rolling two dice and only taking the value of the higher dice roll?
- What can we actually do with congruence relations, specifically?
- $\sqrt{A(ABCD)} =\sqrt{A(ABE)}+ \sqrt{A(CDE)}$
- Conditional Probability and Division by Zero
- Give an equational proof $ \vdash (\forall x)(A \rightarrow B) \equiv ((\exists x) A) \rightarrow B$
- Does the square of uniform distribution have density function?
- When are nonintersecting finite degree field extensions linearly disjoint?
- Normal Random Variable
- If $a+b+c=6$ and $a,b,c$ belongs to positive reals $\mathbb{R}^+$; then find the minimum value of $\frac{1}{a}+\frac{4}{b}+\frac{9}{c}$ .
- Jobs in industry for pure mathematicians
- Is $\mathbf{Q}(\sqrt{2},\sqrt{3}) = \mathbf{Q}(\sqrt{6})$?
- Are there periodic functions without a smallest period?
- What is this series relating to the residues of the Gamma function?
- Prove Cardinality of Power set of $\mathbb{N}$, i.e $P(\mathbb{N})$ and set of infinite sequences of $0$s & $1$s is equal.
- Extensions: Spectrum