I'm trying to find a Taylor series for g^n, the nth iterate of g, centered at a fixed point p of g.(adsbygoogle = window.adsbygoogle || []).push({});

I know the first few terms:

if g^n(x)=a_0 + a_1 (x-p) + a_2 (x-p)^2 + ..., then

let s=g'(p)

a_0=p

a_1=s^n

a_2=[s^(n-1) (s^n - 1) g''(p)] / (2(s-1)).

It's safe to say that a_3 is as complex as a_2 is compared to a_1; it's not pretty.

I have a feeling that FaĆ” di Bruno's formula (http://mathworld.wolfram.com/FaadiBrunosFormula.html) may be involved but I would prefer to avoid that.

One of my goals is to find a formula in which I can let n be something other than an integer to at least approximate some fractional iterates of a function.

Does anyone know a formula for the kth derivative of the nth iterate of g (assuming its sufficiently well behaved)? Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Kth derivative of the nth iterate

**Physics Forums | Science Articles, Homework Help, Discussion**