Tetration extension for bases between 1 and eta

[JmsNxn]

Sorry for continuing to post more limit formulas but I found another that I do not think has been mentioned before.

\( f^n(x) = \lim_{k\to \infty} f^{-k}(\frac {f'(f^k(x))^n(f^k(x)-f(f^k(x)))+f(f^k(x))-f^k(x)f'(f^k(x))} {1-f'(f^k(x))}) \)

which is the same as

\( f^n(x) = \lim_{k\to \infty}f^{-k}(\frac {f'(u)^n(u-f(u))+f(u)-uf'(u)} {1-f'(u)})\\where\\u=f^k(x) \)


This works whenever a function has a regular attracting or repelling fixed point that it increases through. For it to work near a repelling fixed point you simply let k approach negative infinity.

This formula is known to produce Schröder iteration. The proof of that is at https://math.eretrandre.org/tetrationfor...6#pid10036.

where is the proof ? 

copy the relevant part here ?

regards

tommy1729

Actually, you can see that Bo already showed it (he pointed out it's Kneser's iteration formula; i.e: Schroder iteration). I just gave a rough outline of how to prove it. It's pretty obvious if you think about it. The period in \(n\) will be that of the Schroder iteration, which is enough to conclude that it is Schroder's iteration (only one iteration has that period). I didn't exactly prove this, I proved if it converges it converges to Schroder's iteration. And even then, I mostly just sketched it. I also used the Ramanujan formula to show they must be the same too, but again, it was just a sketch.