Extension of tetration to other branches

[mike3]

Actually it does seem to converge. The problem is that it seems to converge to the same value for every z in such cases. I.e., converging to a constant function.

Interesting!

Hmm. Sounds like it's time for a graph... I'll see if I can prepare a 2D one looking at the values of various branches on the real axis (can't do 3D with anything I've got).

There are uncountably many such limit values, yet as constant functions they are "analytically incompatible" (is that a real term?) with the function (you can't analytically continue a constant function to tetration!)

Well, each constant fixed point of b^x is a tetration! I.e. it satisfies c(z+1)=b^c(z).
Does it converge to fixed points?

But it's a constant function, so it cannot be interpreted as analytic continuation of the specific function to another branch (think about the problem in "reverse": how would you analytically continue from this constant function to a non-constant one? You can't). And not all branches of satisfy . Consider the branch . We get . Note that it takes us back to the principal branch. The equation seems to only hold for all when using the principal branch (though there may be some freedom in the choice of cut of course), if we are interpreting the symbol as a specific single-valued branch. As you can see above, though, if we interpret it in a multivalued sense, that values on some branches of equal for values on some branches of , then it is true, as can be seen from the example I just showed. The situation is similar to that with and : for some branch of log for any given but which branch that is will depend on what is. "Multivalued functions" are funny things, you know?