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Infinite tetration of the imaginary unit
#41
(06/24/2011, 04:36 PM)sheldonison Wrote: Update, one Thesis paper I started to read, written by Edgar Arturo Saenz Maldonado on the Brjuno number seems to have the formulas.


.
And .... the formal power series of h {the Seigel disc function} is given by

If h is the solution of the functional equation ... , the coefficients of the series must satisfy (formally) the following recursive relation:
=1, for n=1, and for n>=2,

where in the second summation,

"... By the formulas in question, it is possible to determine the coefficients of the formal power series of ; the denominators of these coefficients can be written as products of the form , for n>=2, since is an irrational number these products could be very small....", which is where the Brjuno number comes from.

The function h is the (inverse of the) Schöder function of f. Its well-known that the case of multiplier , irrational, behaves similar to hyperbolic fixpoints (i.e. ).
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