A nice series for b^^h , base sqrt(2), by diagonalization Gottfried Ultimate Fellow Posts: 889 Threads: 130 Joined: Aug 2007 06/11/2009, 08:36 PM (This post was last modified: 06/12/2009, 03:53 AM by Gottfried.) (06/11/2009, 06:26 PM)Gottfried Wrote: Don't know yet, whether this has some benefit so far. It looks, as if we had a discussion of that recently in Upper superexponential I'm excerpting a bit of Henryk's post: (03/29/2009, 11:23 AM)bo198214 Wrote: As it is well-known we have for $b the regular superexponential at the lower fixed point. This can be obtained by computing the Schroeder function at the fixed point $a$ of $F(x)=b^x$. (...) Now the upper regular superexponential $\operatorname{usexp}$ is the one obtained at the upper fixed point of $b^x$. For this function we have however always $\operatorname{usexp}(x)>a$, so the condition $\operatorname{usexp}(0)=1$ can not be met. Instead we normalize it by $\operatorname{usexp}(0)=a+1$, which gives the formula: (*1) $\hspace{48} \operatorname{usexp}_b(t)=a+\chi^{-1}\left(\ln(a)^x \chi(1)\right)$ (...) My construction in the previous post was obviously the same as that above construction (*1) ... Gottfried (I added the comments //... ) Gottfried Wrote:Then we can write $ \hspace{48} C = \sum_{j=0}^{\infty} c_j*(-1/2)^j \hspace{96} \text{//this is Schroeder-function\ }\chi_2(x) \text{\ for \ } 2^x - 1 \text{\ at \ } x=-\frac12} \\ \hspace{48} \exp^{\circ h}_{\sqrt{2}}(1) =2 + \sum_{k=1}^{\infty} 2* C^k * d_k * v^k \hspace{24} // = 2+2*\chi_2^{-1}(u^h*\chi_2(-1/2)) \\$ and the k'th coefficient in my first mail is just 2* C^k * d_k in the formula above. where the fixpoint "a" is simply given as constant 2 and could be generalized to the symbol. The sum-expression describes the inverse of the schrÃ¶der-function chi^-1 in Henryk's post. The formula for the repelling fixpoint replaces simply 2 by 4 and (1/2-1) by (5/4-1) and uses the adapted schrÃ¶der-function. So I think it's useful to redirect replies to the other thread... Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread A nice series for b^^h , base sqrt(2), by diagonalization - by Gottfried - 03/05/2009, 04:30 PM RE: A nice series for b^^h , base sqrt(2), by diagonalization - by Gottfried - 03/08/2009, 03:41 PM RE: A nice series for b^^h , base sqrt(2), by diagonalization - by Gottfried - 03/08/2009, 09:59 PM Logarithmic behaviour of the super exponential at -2 - by bo198214 - 03/09/2009, 01:28 AM RE: Logarithmic behaviour of the super exponential at -2 - by Gottfried - 03/10/2009, 08:01 AM RE: Logarithmic behaviour of the super exponential at -2 - by bo198214 - 03/10/2009, 02:20 PM RE: Logarithmic behaviour of the super exponential at -2 - by Gottfried - 03/10/2009, 03:24 PM RE: Logarithmic behaviour of the super exponential at -2 - by bo198214 - 03/10/2009, 04:55 PM RE: Logarithmic behaviour of the super exponential at -2 - by Gottfried - 03/10/2009, 07:00 PM RE: Logarithmic behaviour of the super exponential at -2 - by bo198214 - 03/11/2009, 12:24 AM RE: Logarithmic behaviour of the super exponential at -2 - by Gottfried - 03/11/2009, 02:09 AM RE: Logarithmic behaviour of the super exponential at -2 - by bo198214 - 03/11/2009, 10:18 AM RE: Logarithmic behaviour of the super exponential at -2 - by Gottfried - 03/14/2009, 08:16 AM RE: Logarithmic behaviour of the super exponential at -2 - by andydude - 03/31/2009, 05:12 AM RE: A nice series for b^^h , base sqrt(2), by diagonalization - by Ivars - 03/10/2009, 09:09 PM RE: A nice series for b^^h , base sqrt(2), by diagonalization - by andydude - 04/06/2009, 10:23 PM RE: A nice series for b^^h , base sqrt(2), by diagonalization - by Gottfried - 06/10/2009, 02:09 PM RE: A nice series for b^^h , base sqrt(2), by diagonalization - by Gottfried - 06/11/2009, 06:26 PM RE: A nice series for b^^h , base sqrt(2), by diagonalization - by Gottfried - 06/11/2009, 08:36 PM RE: A nice series for b^^h , base sqrt(2), by diagonalization - by Gottfried - 06/10/2009, 03:27 PM

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