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I've seen some of you talking about half-iterates of the sin function and I can only guess this is from a general formulae about partial iterates of polynomial functions. I was wondering if anyone had a link to anywhere explaining how to take the half iterate of any polynomial? I've tried searching all over google but I'm not having any luck. Any help would be greatly appreciated. Thank you.

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(12/14/2010, 03:12 PM)JmsNxn Wrote: I've seen some of you talking about half-iterates of the sin function and I can only guess this is from a general formulae about partial iterates of polynomial functions. I was wondering if anyone had a link to anywhere explaining how to take the half iterate of any polynomial? I've tried searching all over google but I'm not having any luck. Any help would be greatly appreciated. Thank you.

Hm, perhaps -if not too elementary- you could look at

http://go.helms-net.de/math/tetdocs/Cont...ration.pdf
or

http://go.helms-net.de/math/tetdocs/FracIterAltGeom.htm
and some similar treatizes at my webspace. Also we had some discussions / elaborations (??? what a word) here on polynomial iterations.

But perhaps your question is a bit more specific?

And ehm, ps:another "perhaps" - you might reconsider to provide a name because I'd like it more in forums like this to be able to adress msgs more personally.

Gottfried

Gottfried Helms, Kassel

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12/15/2010, 02:40 PM
(This post was last modified: 12/15/2010, 02:41 PM by JmsNxn.)
(12/14/2010, 06:01 PM)Gottfried Wrote: (12/14/2010, 03:12 PM)JmsNxn Wrote: I've seen some of you talking about half-iterates of the sin function and I can only guess this is from a general formulae about partial iterates of polynomial functions. I was wondering if anyone had a link to anywhere explaining how to take the half iterate of any polynomial? I've tried searching all over google but I'm not having any luck. Any help would be greatly appreciated. Thank you.

Hm, perhaps -if not too elementary- you could look at

http://go.helms-net.de/math/tetdocs/Cont...ration.pdf

or

http://go.helms-net.de/math/tetdocs/FracIterAltGeom.htm

and some similar treatizes at my webspace. Also we had some discussions / elaborations (??? what a word) here on polynomial iterations.

But perhaps your question is a bit more specific?

And ehm, ps:another "perhaps" - you might reconsider to provide a name because I'd like it more in forums like this to be able to adress msgs more personally.

Gottfried

My name's James Nixon

And thank you for the links.

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12/16/2010, 03:00 AM
(This post was last modified: 12/16/2010, 03:01 AM by bo198214.)
(12/14/2010, 03:12 PM)JmsNxn Wrote: I've seen some of you talking about half-iterates of the sin function and I can only guess this is from a general formulae about partial iterates of polynomial functions.

the idea is too take half iterates of formal powerseries (not just polynomials).

There is a formula for the composition of formal powerseries and you solve then something like \( f\circ f = F \), comparing and hence determining the coefficients of the formal powerseries. This works only for \( F(0)=0 \), i.e. \( F \) having a fixed point at 0.

In certain cases (e.g. \( F'(0)=1 \) so called parabolic fixed point at 0, e.g. \( e^x-1 \) and \( \sin(x) \)) the resulting powerseries is however not converging. Then you need to take resort in other methods. Perhaps first look at the FAQ, to find in the "General Discussion and Questions". And next take a look at

http://mathoverflow.net/questions/4347/f...5227#45227
I also read your other post, it looks very interesting, however for the next weeks I dont have time to dive into. I hope some other forum members reply.

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(12/16/2010, 03:00 AM)bo198214 Wrote: (12/14/2010, 03:12 PM)JmsNxn Wrote: I've seen some of you talking about half-iterates of the sin function and I can only guess this is from a general formulae about partial iterates of polynomial functions.

the idea is too take half iterates of formal powerseries (not just polynomials).

There is a formula for the composition of formal powerseries and you solve then something like \( f\circ f = F \), comparing and hence determining the coefficients of the formal powerseries. This works only for \( F(0)=0 \), i.e. \( F \) having a fixed point at 0.

In certain cases (e.g. \( F'(0)=1 \) so called parabolic fixed point at 0, e.g. \( e^x-1 \) and \( \sin(x) \)) the resulting powerseries is however not converging. Then you need to take resort in other methods. Perhaps first look at the FAQ, to find in the "General Discussion and Questions". And next take a look at

http://mathoverflow.net/questions/4347/f...5227#45227

I also read your other post, it looks very interesting, however for the next weeks I dont have time to dive into. I hope some other forum members reply.

Thank you for the link. I said polynomials because I thought there would be a general formula which would extend to power series.

I hope someone replies to my other post as well.