complex iteration

[jaydfox]

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.

[Gottfried]

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

I'll coin approximations for some complex and irrational values:

Code:

:
    f: f(x)= exp(x)-1
    z = s'th iterate of f(1)

Code:

Approximations by Euler-summation
   s= I      z ~ 0.847025032777 + 0.373003870989*I
   s=-1      z ~ 0.693147180560 + 0             *I   (log(2))
   s=-I      z ~ 0.847025032777 - 0.373003870989*I
   s= 1      z ~ 1.71828182846  + 0             *I   (exp(1)-1)
   s=1/2+I   z ~ 0.996846811341 + 0.546746701133*I
   s=sqrt(2) z ~ 2.37133411

64 coefficients, approximation by Euler-summation of
different orders (also separately for real and imaginary part)


Gottfried

[jaydfox]

To illustrate why I think fractional iteration has a 0 convergence radius, take the following complex number and iterate it through f(z) = e^z-1:

Code:

Format = <real, imaginary>
<2.38450723099340379456484596525E-2, 9.30416399142415411721600587028E-5>

It should be quite entertaining, and illustrates a point about how does one find fractional iterations once we start including complex numbers.

[jaydfox]

The point is, fractional iteration for reals should be well-defined, so long as we limit the domain and range to reals. We should be able to prove the function exists and can be approximated with arbitrary precision. We might not, however, be able to find a power series which converges if we consider all terms in the series. But this doesn't prevent us from solving tetration regardless.

[Gottfried]

Jay,
I'm just rereading several posts of earlier threads, and came across this one.

The point is, fractional iteration for reals should be well-defined, so long as we limit the domain and range to reals. We should be able to prove the function exists and can be approximated with arbitrary precision. We might not, however, be able to find a power series which converges if we consider all terms in the series. But this doesn't prevent us from solving tetration regardless.

To illustrate the problem for the exp(x)-1 iteration I reactivated some older formulae and computed a table of coefficients which serve for a bivariate powerseries in x and h (the top- and iteration-parameter in the exp(x)-1 iteration). I think, that occuring divergences are not principally unsuited for numerical evaluation, as Euler-summation of some divergent series shows.
I just tried to reverse the summation in the bivariate powerseries using naive Euler-summation of terms, and get a reasonable result and also more insight in the character of divergence...
See
Coefficients for U-tetration

Gottfried