Green Eggs and HAM: Tetration for ALL bases, real and complex, now possible?

[mike3]

How about looking for a function that maps L to 1, and L* to -1? Call this first function "t", since its just a temporary step.
\( t(z) = \frac{(z-\Re(L))}{i\Im(L)} \)

Now, if t(z)=+/-1, then 1-t(z)^2=0, so 1-t(L)^2=0 and 1-t(L*)^2=0. So this might be your function, which maps L to 0 and L* to 0, and does so in a nice analytic function.
\( f(z) = 1-t(z)^2 = (\frac{(z-\Re(L))}{i\Im(L)})^2-1 \)
\( f(z) \approx 0.5592217758698x^2 - 0.3558121306015x - 1.056597524339 \)

Here is a graph of f(sexp(z)) at z=0, from -5i to +5i For all bases with a pair of complex fixed points, the function should exponentially decay to 0 as z goes to \( \pm \Im \infty \). I would think f(z) would have an infinite convolution/Fourier representation.


This does not work for real bases, b<=exp(1/e).

Hmm. It might work. However the function will not be complex-periodic, and therefore will not have a Fourier/exponential series expansion. It needs the periodicity in order to work.