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 One very important formula Ansus Junior Fellow Posts: 20 Threads: 5 Joined: Aug 2008 11/03/2010, 12:21 AM (This post was last modified: 11/03/2010, 02:26 AM by Ansus.) Anybody shell know this very important formula: $\operatorname{sexp}_b(x)=r+\sum _{n=1}^{\infty} \frac{\left(\ln b \right)^{n-1}\left(\ln \left(b^r\right)\right)^{n x}\left(1-r)^n B_n^x}{n!}$ where $B_n^x$ are the Bell's numbers of x-th order and $r=\frac{W(-\log (b))}{\log (b)}$. For integer x one can find them here: http://www.research.att.com/~njas/sequences/A111672 http://www.research.att.com/~njas/sequences/A144150 http://www.research.att.com/~njas/sequences/A153277 This formula can be easily derived from regular teration, but has a long history dating from 1945 ( J. Ginsburg, Iterated exponentials, Scripta Math. 11 (1945), 340-353.) It is notable that tetration and Bell's polynomials of n-th order have applications in quantum physics: http://arxiv.org/abs/0812.4047 « Next Oldest | Next Newest »

 Messages In This Thread One very important formula - by Ansus - 11/03/2010, 12:21 AM RE: One very important formula - by mike3 - 11/03/2010, 12:35 AM RE: One very important formula - by Ansus - 11/03/2010, 01:03 AM RE: One very important formula - by bo198214 - 11/03/2010, 01:03 AM RE: One very important formula - by Ansus - 11/03/2010, 02:27 AM RE: One very important formula - by mike3 - 11/03/2010, 03:16 AM RE: One very important formula - by Ansus - 11/03/2010, 04:21 AM RE: One very important formula - by Daniel - 11/03/2010, 03:03 AM

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