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

[sheldonison]

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where \( b_n \) are properly-chosen "basis functions". I have thought about the Kneser-mapping solution (i.e. regular iteration warped with theta mapping) as a possible set of basis functions... but the problem is this only covers half of the plane (as given, the upper half-plane), and the Cauchy equations require both halves of the plane.
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Do you, perhaps, have any ideas as to how this could be done? The form of solution need not converge on the entire plane, only on and perhaps near the imaginary axis.

You could try mapping the two unit infinite strip to the unit circle using,
\( f(z) = \frac{4}{\pi}\tan^{-1}(z) \)
Here, f(z) maps the unit circle to an infinite strip, from -1 to +1. The left side of the unit circle is mapped to -1+iz, and the right side of the circle is mapped to 1+iz, where iz varies from \( +/-\Im\infty \) I don't know if that would help or not, or whether the singularity at +/-I would be fairly mild, or not, given that the tetration solution also converges to a fixed point at \( +/-\Im\infty \).
- Sheldon