08/14/2007, 05:33 PM

I'm starting to think the problem is complex iteration. If you take the iterating function as a function in two variables, n and z, then its radius of convergence is limited by both variables. We can fix z, and explore the radius of convergence for n.

I've been looking at this problem for days now, taking limits as z goes to zero, and I find absolutely no reason the function cannot be well-defined and computable to arbitrary precision.

However, when you take the series expansion with fixed small z (e.g., 0.001), it seems that everything is well-defined for real n. However, with complex iteration count, you are necessarily pushed off the real line. If you then use integer iteration counts and push the number back into vicinity of 1, the non-real part of the iterate eventually grows non-trivial, forcing the function to start to spiral around z axis. Now, if we try to apply the inverse complex iteration at this point, we should get back to the real line. e^z-1 is entire, so this should in principle be possible, but if it isn't, then this could explain the singularities. If you cannot go safely use an iteration count of 0.0001i, then you cannot safely use an iteration count of 0.0001.

So why does it converge for integer n? Because, the 2nd iteration at z=0.001 is the same as the 1st iteration at f(0.001), which is the same as the zeroeth iteration at f(f(0.001)). Using the zeroeth iteration, you have a zero radius of convergence, but you're using n=0, so there is no problem.

I'll need to investigate this more, but non-complex iteration counts would seem the be the source of the otherwise unseen singularities.

I've been looking at this problem for days now, taking limits as z goes to zero, and I find absolutely no reason the function cannot be well-defined and computable to arbitrary precision.

However, when you take the series expansion with fixed small z (e.g., 0.001), it seems that everything is well-defined for real n. However, with complex iteration count, you are necessarily pushed off the real line. If you then use integer iteration counts and push the number back into vicinity of 1, the non-real part of the iterate eventually grows non-trivial, forcing the function to start to spiral around z axis. Now, if we try to apply the inverse complex iteration at this point, we should get back to the real line. e^z-1 is entire, so this should in principle be possible, but if it isn't, then this could explain the singularities. If you cannot go safely use an iteration count of 0.0001i, then you cannot safely use an iteration count of 0.0001.

So why does it converge for integer n? Because, the 2nd iteration at z=0.001 is the same as the 1st iteration at f(0.001), which is the same as the zeroeth iteration at f(f(0.001)). Using the zeroeth iteration, you have a zero radius of convergence, but you're using n=0, so there is no problem.

I'll need to investigate this more, but non-complex iteration counts would seem the be the source of the otherwise unseen singularities.

~ Jay Daniel Fox