Laplace transform of tetration - 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: Laplace transform of tetration (/showthread.php?tid=298) Laplace transform of tetration - BenStandeven - 06/01/2009 Let a be a base $1 < a < \e^{1/\e}$. Then we can build a regular tetration function $tet_a$ around either of the fixed points. In either case, the function will be periodic, with period given by $per(a) = 2 \pi \i / \log(\log(f))$ for f the fixed point. Thus the Laplace transform of sexp_a is: $tet_a(z) = \sum_{k \in Z} \e^{per(a) k z} c_k$ Here, if we expand around the lower fixed point, all the positive coefficients will be zero, since the function tends to the fixed point at $+\inf$. Similarly, if we expand around the upper fixed point, all the negative coefficients will be zero. In either case, $c_0$ is the chosen fixed point. Now from the equation above, we have $tet_a(z+1) = \sum_{k \in Z} \e^{per(a) k z} [\e^{per(a) k} c_k]$. But by definition, this is equal to $\exp_a(tet_a(z)) = \sum_{n \in N} \frac{$$\sum_{k \in Z} e^{per(a) k z} c_k \log(a)$$^n}{n!}$. By equating the terms of the resulting Laplace series, we get the equation $c_k e^{per(a) k} = \sum_{n \in N} \frac{(\log a)^n}{n!} \sum_{\Sigma k_i = k} $\prod_{i=1}^n c_{k_i}$$. The inner sum is over all integer sequences of length n which sum to k. The finitude of this sum is ensured by the fact that either all positive or all negative coefficients are zero. RE: Laplace transform of tetration - bo198214 - 06/01/2009 (06/01/2009, 06:14 PM)BenStandeven Wrote: Thus the Laplace transform of sexp_a is:Isnt that the Fourier deveopment? Quote:By equating the terms of the resulting Laplace series, we get the equation $c_k e^{per(a) k} = \sum_{n \in N} \frac{(\log a)^n}{n!} \sum_{\Sigma k_i = k} $\prod_{i=1}^n c_{k_i}$$. The inner sum is over all integer sequences of length n which sum to k. The finitude of this sum is ensured by the fact that either all positive or all negative coefficients are zero. And actually the $c_k$ are the coefficients of the inverse SchrÃ¶der powerseries. Incidentally Dmitrii and I just finished an article about exactly that topic, which I append. RE: Laplace transform of tetration - andydude - 06/01/2009 Wow, nice article! I wept. I think one of the parts that was new to me was the proof that the tetrations developed at the fixed points 2 and 4 are different. You show that their periods are different, thus they must be different. So simple!