09/26/2014, 04:47 PM

(09/17/2014, 11:10 PM)jaydfox Wrote:(09/17/2014, 04:17 PM)MorgothV8 Wrote: No one can help how to find all fixed points of log(b), for all complex bases b.

I know exp() have infinite numbe rof fixed points, but two of them are "main" and are the same as log(), I want to find them using some algorithm and elementary functions. I know there is no closed for for them, but I want to find algorithm that approximates them for all bases.

Any ideas?

My program ffrom 1st post inds fixed point - but only one, how to find another one?

(...) if you can use PARI/gp, the following code might be useful to you. (Or to anyone else, like Sheldon?)

So the code I posted previously will generate a Taylor series. I don't know for sure if the Taylor series is entire, or if it has a finite radius of convergence. So far I've managed to calculate the first 1024 terms (took several hours), and the root test sure looks like there's a radius of about 2.5.

The root test climbs up to about 2.77 by the 44th term, then begins descending (in a very cool pattern which I'll explore later).

The 77th term is the last term with a root test over 2.70

The 125th term is the last term with a root test over 2.65

The 233rd term is the last term with a root test over 2.60

I don't have the larger data set (1024th degree), but eyeballing a graph I made, it looks like the root test is a little under 2.49 by that point in the Taylor series (terms 1 mod 4, e.g., 1017th term, 1021st term, etc.).

I don't know how long the root test continues to get smaller, before (or if?) it gets bigger. It could be tens of thousands of terms, which is beyond my ability to calculate. Using my algorithm, we'd need a super-computer. So either we need a better algorithm, or we need some other way of approaching the problem.

By the way, it's a shame that the series is effectively limited to a radius of about 2.5. I found that I can find non-primary fixed points using the same formula. For example, the primary fixed points of base e are 0.3181315 +/- 1.3372357*I. Well, when I plugged sqrt((log(log(e)) + 2*Pi*I)+1) into the formula with 320 terms, I got a value of 2.064031 - 7.59095*I, which isn't too far from a fixed point, 2.06227773 - 7.58863118*I. I assume with more terms it will be even closer to that fixed point. (My 1024-term solution is temporarily unavailable, so I can't test with it at the moment.)

So my working hypothesis is that if we had the full Taylor series, we could calculate primary and at least some (if not all) of the non-primary fixed points for any base (other than 1 or 0). Since the full Taylor series is out of reach, we might look at the idea of analytically extending the function and then developing the Taylor series somewhere that has a larger effective radius (as indicated by the root test).

~ Jay Daniel Fox