andydude Wrote:Here is the power series that corresponds to regular iteration:Hmm, now I see first time the difference... (why not start one thread "Basics" and add (or reference) posts to these basic questions - it may be easier to add information than to add to a mega-document Tex-Faq)
\(
f^t(x)
\ =\ \sum_{k=0}^{\infty} x^k G_k(t)
\ =\ f^t(0)
\ +\ x \left[D_x f^t (x)\right]_{x=0}
\ +\ \frac{x^2}{2} \left[D_x^2 f^t (x)\right]_{x=0}
\ +\ \cdots
\)
And here is the power series that corresponds to natural iteration:
\(
f^t(x)
\ =\ \sum_{k=0}^{\infty} t^k H_k(x)
\ =\ f^0(x)
\ +\ t \left[D_t f^t (x)\right]_{t=0}
\ +\ \frac{t^2}{2} \left[D_t^2 f^t (x)\right]_{t=0}
\ +\ \cdots
\)
Concerning the second: my coefficients for the eigensystem-based analysis shows, that the series w.r.t height t (or h) have t in the exponent; they are *not* powerseries (except for one set of bases), so I wonder, whether the above formal derivative is correct?
[update] hmm, on a second read I may answer this by myself: this difference is coded in the different type of derivatives [D...] only - the taylor formula is true for any type of series.
But there is still one aspect, which I'll think about. Conversion of a zeta-series (the parameter is in the exponent, similar to the expansion of It. dec. exp.) into a representation as a powerseries involves the mystic stieltjes-constants, which are related to the euler-mascheroni-constant gamma. [/update]
Gottfried
Gottfried Helms, Kassel