q concerning derivatives
#1
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


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