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 Iteration basics Gottfried Ultimate Fellow Posts: 789 Threads: 121 Joined: Aug 2007 04/06/2008, 07:16 AM (This post was last modified: 04/06/2008, 07:36 AM by Gottfried.) andydude Wrote:Here is the power series that corresponds to regular iteration: $ 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$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) 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 « Next Oldest | Next Newest »

 Messages In This Thread Iteration basics - by Ivars - 03/20/2008, 10:34 AM RE: Iteration basics - by Ivars - 03/20/2008, 05:24 PM RE: Iteration basics - by Gottfried - 03/21/2008, 08:37 AM RE: Iteration basics - by Ivars - 04/03/2008, 08:05 AM RE: Iteration basics - by Gottfried - 04/03/2008, 08:19 AM RE: Iteration basics - by Ivars - 04/03/2008, 10:13 AM RE: Iteration basics - by andydude - 04/05/2008, 11:35 PM RE: Iteration basics - by Gottfried - 04/06/2008, 07:16 AM RE: Iteration basics - by Ivars - 04/06/2008, 08:55 AM RE: Iteration basics - by andydude - 04/07/2008, 12:01 AM RE: Iteration basics - by Ivars - 04/20/2008, 10:15 PM RE: Iteration basics - by bo198214 - 04/21/2008, 08:21 PM RE: Iteration basics - by andydude - 04/22/2008, 05:30 AM RE: Iteration basics - by Ivars - 04/22/2008, 07:02 AM RE: Iteration basics - by Ivars - 04/09/2008, 07:56 PM RE: Iteration basics - by Ivars - 05/09/2008, 09:45 AM RE: Iteration basics - by bo198214 - 05/09/2008, 02:49 PM RE: Iteration basics - by Ivars - 05/27/2008, 10:33 AM RE: Iteration basics - by Gottfried - 05/27/2008, 07:35 PM RE: Iteration basics - by Ivars - 05/30/2008, 06:08 AM RE: Iteration basics - by Xorter - 01/02/2017, 05:21 PM RE: Imaginary iterates of exponentiation - by Gottfried - 03/20/2008, 12:04 PM RE: Imaginary iterates of exponentiation - by Ivars - 03/20/2008, 12:35 PM RE: Imaginary iterates of exponentiation - by bo198214 - 03/20/2008, 03:10 PM RE: Imaginary iterates of exponentiation - by Ivars - 05/24/2008, 08:35 AM RE: Imaginary iterates of exponentiation - by bo198214 - 05/24/2008, 09:04 AM RE: Imaginary iterates of exponentiation - by Gottfried - 05/24/2008, 09:47 AM RE: Imaginary iterates of exponentiation - by Ivars - 05/24/2008, 07:12 PM

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