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 Inverse super-composition Xorter Fellow Posts: 90 Threads: 29 Joined: Aug 2016 12/25/2016, 10:23 PM (12/25/2016, 08:35 PM)sheldonison Wrote: For a fixed point of zero, with a fixed point multiplier of 2, the general solution for the Abel function generated at the fixed point of zero is: $f(z) = 2x + \sum_{n=2}^{\infty}a_n z^n$ $\alpha(z) = \log_2(S(z))\;\;\;$ This is the Abel function for f(z) $\;\alpha(f(z)) = \alpha(z)+1$ where S(z) is the formal Schröder equation solution; $S(f(z)) = 2\cdot S(z)\;\;\; S(z)=z+\sum_{n=2}^{\infty}b_n z^n\;\;$ This is sometimes called Koenig's solution. It can be modified to work with any fixed point multiplier of k, |k|<>1. Using pari-gp one can easily write a program to generate the formal power series for S(x) given f(x). I almost undestand it. Okey, than what is the a[n] and b[n] in the sum formula? I feel we are closer then ever before. Could you show me this way with another example, please? For instance, let us invastigate it: $cos ^o ^N (x) = sin(x)$. (And cos' fixed point is ~0.739.) The question is that what N is and how I can calculate it. Xorter Unizo « Next Oldest | Next Newest »

 Messages In This Thread Inverse super-composition - by Xorter - 11/24/2016, 12:53 PM RE: Inverse super-composition - by JmsNxn - 11/25/2016, 08:55 PM RE: Inverse super-composition - by Xorter - 12/23/2016, 01:33 PM RE: Inverse super-composition - by JmsNxn - 12/23/2016, 08:12 PM RE: Inverse super-composition - by Xorter - 12/24/2016, 09:53 PM RE: Inverse super-composition - by sheldonison - 12/25/2016, 04:16 AM RE: Inverse super-composition - by Xorter - 12/25/2016, 04:38 PM RE: Inverse super-composition - by sheldonison - 12/25/2016, 08:35 PM RE: Inverse super-composition - by Xorter - 12/25/2016, 10:23 PM RE: Inverse super-composition - by sheldonison - 12/26/2016, 07:10 AM RE: Inverse super-composition - by Xorter - 01/12/2017, 04:19 PM RE: Inverse super-composition - by Xorter - 05/26/2018, 12:00 AM

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