# Tetration Forum

Full Version: Laplace transform of tetration
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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.
(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.
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!