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Mar
12
revised Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
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Mar
12
revised Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
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Mar
12
revised Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
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Mar
12
comment Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
@Will: I've added the image with the half and the one-third-iterate. Later today I'll add a comparision of the Écalle's Abel-function and the "usual" version which is computed by the simple logarithmic iterate - curious, whether we find a difference...
Mar
12
revised Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
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Mar
12
revised Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
added 231 characters in body
Mar
11
revised Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
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Mar
11
revised Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
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Mar
11
revised Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
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Mar
11
revised Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
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Mar
11
comment Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
Will, I did the example-computations using Pari/GP with rational arithmetic and have now 509 coefficients for the Laurent-series of the Abel-function. About 1500 digits in numerator as well as in the denominator for the coefficient at the last nonzero term! The series has zero range of convergence in the same way as the "regular" fractional iterates of the $\exp(x)-1$ (that was a surprise to me, I thought the Écalle-method had a series with nonzero range of convergence.... Please see details in my new answer.
Mar
11
answered Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
Mar
9
comment Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
:-) I'll tell you after the soccer game... :-)
Mar
9
comment Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
Ah, thanks! Well I use Pari/GP for all this, and it is fast enough for 200 decimal digits precision as default!
Mar
9
comment Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
Ah well that's enough to say. I can do the bisection myself. I only thought with such a basic thing like the Abel-function the inverse were something similarly well developed and I were just missing the info. Thank you!
Mar
9
comment Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
I forgot to ask when I read this first time but want to implement it to experiment. How does this recipe compute the inverse of the $\alpha()$ ? (Or is it somehow trivial?)
Mar
1
revised Is there an “elegant” non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?
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Mar
1
revised Is there an “elegant” non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?
I had worked for the original schröderfunction instead for the inverse. Corrected/text adapted
Mar
1
revised Is there an “elegant” non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?
I had worked for the original schröderfunction instead for the inverse. Corrected/text adapted
Mar
1
revised Existence of a square root of a functional equation
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