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 q concerning derivatives Gottfried Ultimate Fellow Posts: 757 Threads: 116 Joined: Aug 2007 03/19/2008, 08:28 AM (This post was last modified: 03/19/2008, 09:22 AM by Gottfried.) Comparing the derivations for formulae for fractional iteration using derivatives vs. fractional powers of matrices, I came to a consideration, which took me last days, but is still vague for me. I recall, that the eigensystem-decomposition provides a matrix of coefficients, which allow to write the general iteration of height h and "start-parameter" x, say $E_1(x)=t^x -1$, and $E_1^{o h}(x)$ the h'th iterate as function of two parameters (using u=log(t)) and $f_k(x)$ are different (powerseries-defined) functions of x, also dependent on u: $ E_1^{o h}(x) = a_0*f_0(x)*u^h + a_1*f_1(x)*u^{2h} + a_2*f_2(x)*u^{3h} + ...$ for instance u=2 $ E_1^{o h}(x) = a_0*f_0(x)*2^h + a_1*f_1(x)*4^h + a_2*f_2(x)*8^h + ...$ where, if we derive wrt x we use derivatives according to derivation of powerseries; but if we derive wrt h, we have to use derivation of dirichlet-like series, which is quite different. Well, if I fix x here, denote $E_1^{o h}(x_0) = F(h)$ and write the Euler-Maclaurin-formula for the sum of derivatives $F^{(k)}(0)/k!$ expand and sum, I get formally the correct result for F(1). So basically I don't have any problem here (besides that I'll have to become more familiar with this), but since I didn't see this difference mentioned anywhere, I wonder, whether I've it simply overlooked or whether it is even trivial... [update] Well, it's not that I didn't understand derivatives, it's only, that I was extremely focused on my matrix-concept, and now get aware of some more very simple relations. So maybe it's only a "whoops"-reminder... [/update] Scratching head... Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

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