11/22/2007, 11:35 PM

To remove any tetration-specific issues from discussion for a moment, let's consider a function , with second iterate , and generally, .

Well, the integer iterates are very well-defined. But what about the fractional iterates? For starters, the rational iterators have a finite number of branches. Sure, for real z>0, there is a well-defined "principal branch", but when taking z to be complex, we must admit that the branches prevent us from having unique solutions.

Not so with the integer iterates, however. No branching. Or from another perspective, an infinite number of branches that are all identical, giving the appearance that there are no branches. From either perspective, there are effecitvely no branches.

Getting back to the non-integer iterates, we find a finite number of branches for rational iterates, and an infinite number of branches for irrational iterates.

I bring this up, because it helps put into perspective something that I had felt uneasy about with the fractional iterates of e^x. As I look more and more at the slog, it seems to heavily favor integer iterates, in many different ways. Yet exponentiation doesn't seem to favor integer iterates of multiplication any more than non-integer iterates, so I had hoped that tetration would not favor integer iterates of exponenitation.

But when viewing the base of multiplication as a variable, it becomes clear that exponentiation does favor integer iterates. And it thus comes as less of a surprise that tetration should favor integer iterates, in certain basic contexts.

Another observation: there are singularities in the slog that appear in only certain branches. This seemed bizarre to me at first, and even after I got used to it, it still seemed hard to relate to other basic functions.

But I found a simple way to generate this same effect, useful for studying how unseen singularities can be predicted, even when they lie within the radius of convergence in a different branch.

Take the following multi-valued function:

When z = 4, the denominator goes to 0, but only in the non-principal branch. You see, at z=4, the principle branch gives us:

However, in the second branch, we get sqrt(4) = -2, which then gives us a singularity. If you develop the power series at z=4 in the principal branch, you'll see the radius of convergence limited by the singularity at z=0. You can then use analytic continuation to verify that there is indeed a singularity at z=4 in the second branch.

The question I have is, can we use a similar technique (constructing a singularity in a non-principal branch) to build the slog from basic functions such as logarithms, exponentiations, etc., without resorting to "cheats" such as Taylor series or Laurent series or Fourier series?

Well, the integer iterates are very well-defined. But what about the fractional iterates? For starters, the rational iterators have a finite number of branches. Sure, for real z>0, there is a well-defined "principal branch", but when taking z to be complex, we must admit that the branches prevent us from having unique solutions.

Not so with the integer iterates, however. No branching. Or from another perspective, an infinite number of branches that are all identical, giving the appearance that there are no branches. From either perspective, there are effecitvely no branches.

Getting back to the non-integer iterates, we find a finite number of branches for rational iterates, and an infinite number of branches for irrational iterates.

I bring this up, because it helps put into perspective something that I had felt uneasy about with the fractional iterates of e^x. As I look more and more at the slog, it seems to heavily favor integer iterates, in many different ways. Yet exponentiation doesn't seem to favor integer iterates of multiplication any more than non-integer iterates, so I had hoped that tetration would not favor integer iterates of exponenitation.

But when viewing the base of multiplication as a variable, it becomes clear that exponentiation does favor integer iterates. And it thus comes as less of a surprise that tetration should favor integer iterates, in certain basic contexts.

Another observation: there are singularities in the slog that appear in only certain branches. This seemed bizarre to me at first, and even after I got used to it, it still seemed hard to relate to other basic functions.

But I found a simple way to generate this same effect, useful for studying how unseen singularities can be predicted, even when they lie within the radius of convergence in a different branch.

Take the following multi-valued function:

When z = 4, the denominator goes to 0, but only in the non-principal branch. You see, at z=4, the principle branch gives us:

However, in the second branch, we get sqrt(4) = -2, which then gives us a singularity. If you develop the power series at z=4 in the principal branch, you'll see the radius of convergence limited by the singularity at z=0. You can then use analytic continuation to verify that there is indeed a singularity at z=4 in the second branch.

The question I have is, can we use a similar technique (constructing a singularity in a non-principal branch) to build the slog from basic functions such as logarithms, exponentiations, etc., without resorting to "cheats" such as Taylor series or Laurent series or Fourier series?