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 2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? Gottfried Ultimate Fellow Posts: 766 Threads: 119 Joined: Aug 2007 09/11/2013, 08:32 PM (This post was last modified: 09/12/2013, 07:28 AM by Gottfried.) ahh, stupid.... Of course the basic Chebychev-polynomials are "the-functions-of which-acosh(x)-is-the-SchrĂ¶derfunction". Let for instance $f_2(x) = T_2(x) = -1 + 2x^2$ then of course the iterates $f_2^{\circ 2}(x) = T_2(T_2(x)) = 1 -8x^2+8x^4$ and the iterates to any height h are $f_2^{\circ h}(x)= T_2^{\circ h}(x)$ which can then be evaluated/interpolated using the cosh/acosh-pair as $f_2^{\circ h}(x) = \cosh(2^h \operatorname{acosh}(x))$ . So my question is then -very simple- answered for "base=2" that the "more common" function "in our toolbox" is the quadratic polynomial $f_2(x)=-1 + 2x^2$ (where the fractional iterates become power series instead...). And if I take any higher-indexed Chebychev-polynomial as the base-function , the iterate is simply computable by the cosh/acosh-mechanism $f_b^{\circ h}(x) = \cosh(b^h \operatorname{acosh}(x))$ (And for my initial question using sinh/asinh it is simply the same except with other polynomials) Should have seen this before... - Gottfried [update]: And -oh wonder- this is just related to a question in MSE where I was involved this days without knowing that this two questions are in the same area; I just added the information about the cosh/arccosh-composition there... :-) http://math.stackexchange.com/questions/...965#490965 Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread 2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? - by Gottfried - 09/10/2013, 12:23 PM RE: 2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? - by tommy1729 - 09/10/2013, 09:50 PM RE: 2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? - by tommy1729 - 09/10/2013, 10:06 PM RE: 2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? - by mike3 - 09/11/2013, 10:49 AM RE: 2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? - by Gottfried - 09/11/2013, 06:30 PM RE: 2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? - by Gottfried - 09/11/2013, 08:32 PM

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