Extension of tetration to other branches

[bo198214]

But it's a constant function, so it cannot be interpreted as analytic continuation of the specific function \( \mathrm{tet}_b(z) \) to another branch

Yes you are right, its no more the regular tetration.
Can you say which branch sequence you used to produce the constant function?

As you can see above, though, if we interpret it in a multivalued sense, that values on some branches of \( \mathrm{tet}_b(z+1) \) equal \( b^{\mathrm{tet}_b(z)} \) for values on some branches of \( \mathrm{tet}_b(z) \), then it is true, ... "Multivalued functions" are funny things, you know?

Indeed I noticed that interpretation already in other contexts. If you have a multivalued function satisfying a certain functional equation it satisfies this equation only if you choose the suitable branches.

Easiest example is log with the functional equation log(ab)=log(a)+log(b).
I remember also this kind of description in [Kuzma: iterative functional equations] for regular holomorphic Abel functions. Particularly I noticed this behaviour with Dmitrii's superlog, and now you add regular tetration.