The "cheta" function

[sheldonison]

My change of base formula relies on the following:
\( \lim_{k \to \infty} \log_a^{\circ k} \left( \exp_b^{\circ k} (x) \right) \).

I had found that it was computationally more accurate to work with the double logarithm of x.....

I've been playing around with this for a few months, on and off. I believe this limit does not converge in the complex plane. First, some background, lets take Jay's forumula, and change base a to e, and base b to eta, and then plug in cheta(x+theta) for x. Here, \( \theta(x) \) is the 1-cyclic base conversion constant that oscillates with a small amplitude around the base conversion "constant". See my previous post for some more details.

\( f(x) =
\lim_{k \to \infty} \log_e^{\circ k} \left( \exp_\eta^{\circ k} (
x
) \right) \)
\( \text{sexp_e(x)} =
\lim_{k \to \infty} \log_e^{\circ k} \left( \exp_\eta^{\circ k} (
\text{cheta}(x+\theta(x))
) \right) \)

The second equation is interesting, especially since cheta is entire. If the theta is replaced with a constant, we get Jay's base change version of the sexp solution. So it would be nice to get a graph of the f(x) limit equation in the complex plane. Here's an overview of my attempt to explain why the f(x) limit equation may not converge in the complex plane. First, show the f(x) limit converges nicely for some particular real value of x. Second, show that f(x+imaginary) converges to a very different number. Third, show that no matter how small the imaginary component is, f(x+imaginary) converges to a very different number than f(x). Then the slope in the complex plane is discontinuous, and perhaps the convergence radius is zero, or f(x) is not analytic. There is one other possibility that I cannot rule out, and that is that the multi-valued logarithm allows for a solution that does converge.

The limit equation for base e, eta, f(x) takes on complex values for x<=4.384, and this is expected. For larger real values of x, the limit is well defined, real valued, and quickly converging. I analyzed f(x=4.7), as k increases. The double log formula makes it fairly easy to reach arbitrarily large precisions. For k=5, the eta power tower is already 8.3*(10^150).
k=0, f(x)=4.7
k=1, f(x)=1.729, shortcut, f(x)=x/e
k=2, f(x)=0.729, shortcut, f(x)=x/e-1, Jay's double log forumula
k=3, f(x)=0.0705
k=4, f(x)=-0.4237
k=5, f(x)=-0.5638
k=6, f(x)=-0.5642928...
k>=7, f(x)=-0.5642928....

Convergence works perfectly for real numbers>0.4385. Now we try complex numbers, x=4.7+0.2i. As near as I can tell, the sequence will approach the fixed point for sexp_e.

k=0, f(x)= 4.7 + 0.2i
k=1, f(x)= 1.7290 + 0.0736i
k=2, f(x)= 0.7290 + 0.0736i
k=3, f(x)= 0.0754 + 0.1418i
k=4, f(x)=-0.3640 + 0.3394i
k=5, f(x)=-0.4191 + 0.4583i
k=6, f(x)=-0.4184 + 0.4574i
k=7, f(x)=-0.0342 + 1.4238i
k=8, f(x)= 0.3094 + 1.6065i
k=9, f(x)= 0.4950 + 1.3975i
k=10,f(x)= 0.4020 + 1.2313i
k=11,f(x)= 0.2605 + 1.2507i
k=12,f(x)= 0.2423 + 1.3639i
k=13,f(x)= 0.3247 + 1.3964i
....

Here is the sequence of exp_eta(x) that leads to f(x) approaching the fixed point of e. Notice that at k=3, the cos(imaginary_component/e) is negative.
k=0, x=4.7 + 0.2i
k=1, eta^x= 5.619958178 + 0.414241171i
k=2 eta^eta^x = 7.813166879 + 1.199958104i
k=3, 16.01506009 + 7.567768883i cos(imag/e) negative
k=4, -339.0928882 + 126.694129i real value negative
k=5, -5.79817E-55 + (3.28697E-55)i real value close to zero
k=6, 1 + (1.20921E-55)i
k=7, 1.444667861 + (6.42651E-56)i
k>=8 grows arbitrarily close to e.

There's nothing special about the starting point of i=0.2. Any non-zero imaginary starting point will eventually lead to the magnitude of the imaginary component of exp_eta growing, until its imaginary component is greater than e*(pi/2). If at that point in time the cos(imag/e)<0, which is a 50/50 proposition, than the real component for the next iteration goes negative. The next iteration after that is very close to zero. And from that point forward, the exp_eta will grow forever towards e. The logarithms are multi-valued so that's trickier, but it would appear that they approach the fixed point of sexp_e.

Moreover, there is no "limit" in the conventional sense. No matter how close the initial imaginary portion approaches zero, eventually the iterated super exponential will grow so that the imag/e>pi/2, and from that point on, its a random roll of the dice. Again, the multi-valued logarithm may allow for a different solution that does converge in the complex plane. However, the increasingly fractal nature of the repeated exponentiation may makes it difficult to generate a continuous solution. Some values of the initial imaginary may arbitrarily lead to all positive values of the cosine(imag/e) at every iteration, and grow forever, and lead to nicely behaved values of f(x). However, most values of the initial imaginary component will eventually lead to a negative component of cosine(imag/e), and will henceforth lead to exp_eta growing towards "e". The combination of the two types of behavior may provide a way to show the limit doesn't converge.
- Sheldon Levenstein