(07/10/2010, 05:17 AM)bo198214 Wrote: Another formula (2.29 in the overview paper) that kinda combines hyperbolic and parabolic is:
\( \lim_{n\to\infty}
\frac{f^{[n]}(v)-f^{[n]}(z)}{f^{[n+1]}(z)-f^{[n]}(z)}=w\frac{1-\lambda^{w}}{1-\lambda}
\), \( f^{[w]}(z)=v \), \( \alpha_z(v)=w \)
where \( \lambda \) is the derivative at the fixed point 0, which is 1 in the parabolic case and you take the limit of lambda->1. I am in a hurry a bit. So perhaps more detailed later.
(07/21/2010, 10:41 PM)tommy1729 Wrote: im waiting and hoping for those " more details " dear bo.
Well, there is no more much to add, if you take the limit of the above right side:
\( \lim_{\lambda\to 1} w\frac{1-\lambda^{w}}{1-\lambda} \)
you get \( w \) if I not err, which is then the parabolic Levý formula.
If you could invert \( h(w)=w\frac{1-\lambda^{w}}{1-\lambda} \) for \( \lambda\neq 1 \) then \( h^{-1}\left(\frac{f^{[n]}(v)-f^{[n]}(z)}{f^{[n+1]}(z)-f^{[n]}(z)}\right) \) would be another formula for the hyperbolic Abel function.
If you can't (numerically/symbolically whatever) invert then you still have a different formula for the hyperbolic (and Lévy's formla for \( \lambda\to 1 \)) superfunction/iteration:
\( f^{[w]}(z)=f^{[-n]}(h(w)*(f^{[n+1]}(z)-f^{[n]}(z)) + f^{[n]}(z)) \)