half operator problem -- wondering if this is a discussed topic
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From: Gottfried Helms on 14 Jun 2010 01:30 Am 11.06.2010 05:05 schrieb Lee Davidson: > > Thanks all. Now, it has occurred to me that one approach would be > through power series, iterated. Supposing > > g(x) = a_0 + a_1 x + a_2 x^2 ... > > Let f(x) = b_0 + b_1 x + b_2 x^2 ... > > Then f(x) = g(g(x)) = a_0 + a_1 g(x) + a_2 g(x)^2 ... > > Then substitute g's power series and f's power series and solve the > simultaneous equations. > > Aaaaarrrrgh! For a basic introduction you might like this: http://go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf I give two examples for powerseries, using matrix-notation for the formal powerseries and use of that matrices as operators, including the use of log resp. the diagonalization of the operators to find powerseries for the fractional iterates. The idea is to find formulae (for instance polynomials) for the coefficients of the powerseries, such that their coefficients for the h'th iteration can be computed by a formula involving h. I deal mainly with easier case where a0=0. For a1=1 I show examples using the log of the matrix-operator and for 0<a1<1 one can use the diagonalization. Also, allowing divergent summation it seems, that 1<a1 is manageable... On p 29 I also step into the a0<>0 case. However, mine is only some amateurish collection. You can find much more material and more sophisticated treatment if you google for "tetration"; in the "tetration-forum" and elsewhere. For the iterated exp-function there is a recent article of H.Trappmann/D.Kouznetsov describing various approaches to an analytic iteration. Gottfried Helms Henryk Trappmann, "5+ methods for real analytic tetration" (see link in the "literature"-thread in tetration-forum http://math.eretrandre.org/tetrationforum/showthread.php?tid=365 ) D.Kouznetsov. (2009). Solutions of F(z+1)=exp(F(z)) in the complex zplane.. Mathematics of Computation, 78: 1647-1670. Dmitrii Kouznetsov/Henryk Trappmann Portrait of the four regular Super-exponentials to base sqrt(2) to appear in "Mathematics of Computation" (or did it already appear?) http://en.citizendium.org/wiki/Tetration (Kouznetsov/Trappmann) |