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 Tetration FAQ Discussion Gottfried Ultimate Fellow     Posts: 767 Threads: 119 Joined: Aug 2007 07/17/2008, 10:48 AM Code:```-- What does half-/ fractional-/continuous-/complex iteration mean? This problem can only be expressed in terms of the series-paradigm, although the Reihenalgebra-concept can possibly be seen as an equivalent approach. The question is - for example -   given a function f(x) = y   what is the function g(x) such that g(g(x)) = y   g(x) is then called the half-iterate of f(x) and is a fractional-iterate    The terms half,fractional and continuous are used if the iterator-parameter is thought as real, but continuous; if the iterator is thought as a general complex number, sometimes the term continuous is as well used.    For real iterator h   f°h(x) = f°(n+r)(x) = f°n(f°r(x))    where n is integer and r is fractional Example using powerseries: For a function f(x), defined by powerseries, with constant term=0 (f(0)=0) and f'(0)=/=0 it is easy to find the half iterate g(x) by manipulation of the powerseries and equating coefficients at like powers of x:   Assume f(x) = Ax + Bx^2 + Cx^3 + ...   target g(x) = ax + bx^2 + cx^3 + ...   satisfying g(g(x)) = f(x)   then      g(g(x)) = a g(x) + b g(x)^2 + c g(x)^3 + ...           =  a*( ax + bx^2 + cx^3 + ...)            + b*( ax + bx^2 + cx^3 + ...)^2            + ...           = a^2 x + (ab + ba^2) x^2 + ...   = f(x)  =  A  x +    B        x^2 + ...   then by equating coefficents at like powers of x , either a=+sqrt(A) or a=-sqrt(A)   and all other coefficients can then uniquely be determined, so we get       g(x) = sqrt(A) x + B/(sqrt(A) + A)*x^2 + ...    For general fractional iteration-heights the handling of the appropriate powerseries is much more complicated and suggests the tools of algebra of infinitely-sized matrices. Formal analytical handling for general functions is much developed and mostly based on   (see:) Abel - functional relation          Schröder - functional relation            [see : matrix-approach, matrix-logarithm, matrix-diagonalization,          binomial-expansion using functions, ~ using matrix-operators,          function-logarithm (ILog) , exponential polynomial interpolation          ]   [see : Faa di Bruno-formula, ... ]   [see further : iteration-theory, time-series, dynamical systems]    ====================================================================================``` Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread Tetration FAQ Discussion - by bo198214 - 08/09/2007, 12:07 AM Literature - by Gottfried - 11/23/2007, 04:30 PM RE: Literature - by andydude - 12/17/2007, 07:31 PM RE: Tetration FAQ - by andydude - 01/13/2008, 08:43 AM RE: Tetration FAQ - by Ivars - 01/13/2008, 10:17 AM RE: Tetration FAQ - by GFR - 01/13/2008, 04:02 PM RE: Tetration FAQ - by andydude - 01/16/2008, 04:13 AM RE: Tetration FAQ Discussion - by Gottfried - 07/17/2008, 10:34 AM FAQ-discuss: What is a fixpoint? - by Gottfried - 07/17/2008, 10:45 AM FAQ-discuss: What does f o g mean? - by Gottfried - 07/17/2008, 10:47 AM FAQ-discuss: What does half-/ fractional-/continuous-/complex iteration mean? - by Gottfried - 07/17/2008, 10:48 AM RE: FAQ-discuss: What does half-/ fractional-/continuous-/complex iteration mean? - by andydude - 07/17/2008, 02:52 PM Intro and Formula sheet - by andydude - 11/13/2007, 01:21 AM RE: Intro and Formula sheet - by jaydfox - 11/13/2007, 02:39 AM RE: Intro and Formula sheet - by andydude - 11/13/2007, 03:15 AM RE: Intro and Formula sheet - by andydude - 11/13/2007, 03:25 AM RE: Intro and Formula sheet - by Gottfried - 11/13/2007, 06:08 AM RE: Intro and Formula sheet - by Gottfried - 11/13/2007, 06:12 AM RE: Intro and Formula sheet - by andydude - 11/13/2007, 08:37 AM RE: Intro and Formula sheet - by andydude - 11/13/2007, 09:09 AM RE: Intro and Formula sheet - by bo198214 - 11/13/2007, 10:42 AM RE: Intro and Formula sheet - by andydude - 11/15/2007, 03:16 AM RE: Intro and Formula sheet - by jaydfox - 11/15/2007, 03:38 AM RE: Intro and Formula sheet - by andydude - 11/30/2007, 05:32 PM [split] Tetration FAQ - by Ivars - 07/11/2008, 06:28 AM

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