As I mentioned in this thread, it appears the analytical continuation of the Kneser tetrational (or a function equivalent to it) into the complex plane looks like a fusion of the regular iterations at two different fixed points (even at the real axis for bases greater than eta it is, but the construction there makes reference to only one due to the conjugate symmetry):

http://math.eretrandre.org/tetrationforu...hp?tid=530

But what happens at base ? A long-standing question has been whether or not the tetrational is analytic here. I think the answer to this is no, and here I give some arguments that I think provide very good evidence for this being true, though not a rigorous proof (that may require a more “explicit” formula (like a Taylor series w/explicit coefficients) for the Kneser tetrational.).

One thing to look for, that might indicate that a point is a singularity, is if the function behaves differently there than at any other point in a significant way. Consider what happens at . The Kneser tetrational approaches the parabolic regular iteration. When we take the corresponding superlog, it is, um, weird and it fails to permit iteration on most of the positive real axis. There's something funny about this base.

Let's consider another investigation. We hypothesize that toward the extended Kneser tetrational always looks like the regular at a fixed point , and at it looks like the iteration at another fixed point , which for real are the principal conjugate fixed points. With that in mind, we see that as we approach , the two points merge into each other. This also suggests that is exceptional, again, indicating that it may be a singularity.

But now for the kicker. Consider moving through the complex plane, starting at a real base . What does this do to the fixed points determining the Kneser function? We have

but is multivalued (as is , too.). This formula then gives all fixed points. Let's look at itself. Then, we have … but is a branch point of !!!! Since appears to be a branch point of . If we choose cuts so that is cut from the above with a cut going from to , and the same for , and so that for they equal the and we expect before, then if we wind around from that axis in both directions, we see the branchpoint at means that the and approach different values as we come around to some base .

If we carry this process out (I did a computational test by slowly moving the base and then correcting and with Newton's method, not by computing directly since I don't have facilities for computing non-principal branches of ), we see that winding from, say, to via the upper half-plane (i.e. going counterclockwise), goes to 2 and to 4. We don't even need to carry out the process in the lower half-plane. Already we see that these are not conjugate, thus the tetrational will not be conjugate-symmetric and hence cannot be real at real heights. This contrasts with the behavior at . If a complex analytic function is real along some part of the real axis, but not another, there must be a branchpoint (not sure what the proof is, though.). The obvious candidate here is... . Indeed, choosing bases closer to and repeating the given procedure suggests this is indeed the case: conjugate and for , not conjugate for .

But if we do go around through the lower half-plane, we get and . Note that this may at first be thought to be the same since these are the same two fixed points as before, it is NOT, since now the is 2, not 4, and the is 4, not 2: they have been swapped! Then, we see that this function cannot be the same. We approach two different functions going in two different directions around , which is the defining quality of a branch point.

So, it seems that a very strong case can be made that the Kneser tetrational is:

1. NOT analytic at

2. multivalued in the base as well as the height

3. NOT equal to the regular iteration for

4. has a branch point at .

Could there be other branch points? Considering as the determining function, we can try out other candidates.

Base 1: Here, the -function is indeterminate, since it reduces to 0/0. Though 0 (log of 1) is not a branch point on the principal branch of , it is in the full multivalued -relation, on every other branch (every non-principal one). Thus base 1 could be a BP, though perhaps not on the “principal branch” of tetrational (i.e. cut from to ).

Base 0: Here, has a branch point. So this is a branch point of . It too, is therefore likely a branch point of the tetrational.

So I think we could suspect that tetration has at least three branch points in the base: 0, 1, and , though they may not all be present on the principal branch.

As an aside, we can also use this to attempt to visualize what the true tetrational at looks like. If we choose the "continuous from above" convention at the cut, then it will look like the familiar attracting regular iteration on the upper half-plane, and like the "upper" iteration on the lower half-plane. It will not decay to a fixed point at , but to two different periodic cycles.

http://math.eretrandre.org/tetrationforu...hp?tid=530

But what happens at base ? A long-standing question has been whether or not the tetrational is analytic here. I think the answer to this is no, and here I give some arguments that I think provide very good evidence for this being true, though not a rigorous proof (that may require a more “explicit” formula (like a Taylor series w/explicit coefficients) for the Kneser tetrational.).

One thing to look for, that might indicate that a point is a singularity, is if the function behaves differently there than at any other point in a significant way. Consider what happens at . The Kneser tetrational approaches the parabolic regular iteration. When we take the corresponding superlog, it is, um, weird and it fails to permit iteration on most of the positive real axis. There's something funny about this base.

Let's consider another investigation. We hypothesize that toward the extended Kneser tetrational always looks like the regular at a fixed point , and at it looks like the iteration at another fixed point , which for real are the principal conjugate fixed points. With that in mind, we see that as we approach , the two points merge into each other. This also suggests that is exceptional, again, indicating that it may be a singularity.

But now for the kicker. Consider moving through the complex plane, starting at a real base . What does this do to the fixed points determining the Kneser function? We have

but is multivalued (as is , too.). This formula then gives all fixed points. Let's look at itself. Then, we have … but is a branch point of !!!! Since appears to be a branch point of . If we choose cuts so that is cut from the above with a cut going from to , and the same for , and so that for they equal the and we expect before, then if we wind around from that axis in both directions, we see the branchpoint at means that the and approach different values as we come around to some base .

If we carry this process out (I did a computational test by slowly moving the base and then correcting and with Newton's method, not by computing directly since I don't have facilities for computing non-principal branches of ), we see that winding from, say, to via the upper half-plane (i.e. going counterclockwise), goes to 2 and to 4. We don't even need to carry out the process in the lower half-plane. Already we see that these are not conjugate, thus the tetrational will not be conjugate-symmetric and hence cannot be real at real heights. This contrasts with the behavior at . If a complex analytic function is real along some part of the real axis, but not another, there must be a branchpoint (not sure what the proof is, though.). The obvious candidate here is... . Indeed, choosing bases closer to and repeating the given procedure suggests this is indeed the case: conjugate and for , not conjugate for .

But if we do go around through the lower half-plane, we get and . Note that this may at first be thought to be the same since these are the same two fixed points as before, it is NOT, since now the is 2, not 4, and the is 4, not 2: they have been swapped! Then, we see that this function cannot be the same. We approach two different functions going in two different directions around , which is the defining quality of a branch point.

So, it seems that a very strong case can be made that the Kneser tetrational is:

1. NOT analytic at

2. multivalued in the base as well as the height

3. NOT equal to the regular iteration for

4. has a branch point at .

Could there be other branch points? Considering as the determining function, we can try out other candidates.

Base 1: Here, the -function is indeterminate, since it reduces to 0/0. Though 0 (log of 1) is not a branch point on the principal branch of , it is in the full multivalued -relation, on every other branch (every non-principal one). Thus base 1 could be a BP, though perhaps not on the “principal branch” of tetrational (i.e. cut from to ).

Base 0: Here, has a branch point. So this is a branch point of . It too, is therefore likely a branch point of the tetrational.

So I think we could suspect that tetration has at least three branch points in the base: 0, 1, and , though they may not all be present on the principal branch.

As an aside, we can also use this to attempt to visualize what the true tetrational at looks like. If we choose the "continuous from above" convention at the cut, then it will look like the familiar attracting regular iteration on the upper half-plane, and like the "upper" iteration on the lower half-plane. It will not decay to a fixed point at , but to two different periodic cycles.