Thread Rating:
• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? Gottfried Ultimate Fellow Posts: 789 Threads: 121 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

 Possibly Related Threads... Thread Author Replies Views Last Post using sinh(x) ? tommy1729 100 217,671 10/03/2021, 10:12 PM Last Post: tommy1729 Some "Theorem" on the generalized superfunction Leo.W 44 11,023 09/24/2021, 04:37 PM Last Post: Leo.W Generalized Kneser superfunction trick (the iterated limit definition) MphLee 25 9,023 05/26/2021, 11:55 PM Last Post: MphLee [exercise] fractional iteration of f(z)= 2*sinh (log(z)) ? Gottfried 4 1,927 03/14/2021, 05:32 PM Last Post: tommy1729 Natural cyclic superfunction tommy1729 3 6,401 12/08/2015, 12:09 AM Last Post: tommy1729 exp^[3/2](x) > sinh^[1/2](exp(x)) ? tommy1729 7 12,608 10/26/2015, 01:07 AM Last Post: tommy1729 [2014] Uniqueness of periodic superfunction tommy1729 0 3,574 11/09/2014, 10:20 PM Last Post: tommy1729 twice a superfunction tommy1729 5 10,258 03/26/2014, 03:34 PM Last Post: MphLee The fermat superfunction tommy1729 3 6,764 03/24/2014, 12:58 AM Last Post: tommy1729 Uniqueness of Ansus' extended sum superfunction bo198214 4 11,253 10/25/2013, 11:27 PM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)