real slog developed at a fixed point  Printable Version + Tetration Forum (https://math.eretrandre.org/tetrationforum) + Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) + Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) + Thread: real slog developed at a fixed point (/showthread.php?tid=215) Pages:
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RE: real slog developed at a fixed point  bo198214  12/30/2008 Kouznetsov Wrote:bo198214 Wrote:Didnt you say that the series does not converge? Or not to the values of slog?Not to the values of slog. The radius of convergence of the subseries of terms with integer poers of seems to be . Oh I just see, the construction I made above is the regular slog. I.e. the function Knesers starts with. Thatswhy it doesnt yield the values of the kslog, especially it is not real at the real axis. It is the powerseries development of the regular Abel function, while Kneser works with the limit formula of the regular Abel/Schroeder function. So the continuation of this series to the upper half plane has a singularity at for each . Here the singularity at 0 limits the convergence radius to . Lets call this super logarithm rslog while we call your super logarithm kslog. Then we know already that or where is 1periodic. By the above consideration we can develop for a powerseries . And every 1periodic holomorphic function can be developed as: Then Now the exponential of a powerseries is again a powerseries: inserted: thatswhy we can finally write: which is slightly more unspecific than Knesers's formula. RE: real slog developed at a fixed point  andydude  04/06/2009 @Henryk Did you forget a binomial coefficient somewhere? More specifically: Quote:Now we know that Andrew Robbins RE: real slog developed at a fixed point  bo198214  04/06/2009 andydude Wrote:@Henryk Yes, you are right. I will correct it in my post. However I think the coefficients are still ok (because I computed them by other means.) 