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Laplace transform of tetration
#1
Let a be a base . Then we can build a regular tetration function around either of the fixed points. In either case, the function will be periodic, with period given by for f the fixed point.

Thus the Laplace transform of sexp_a is:



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 . Similarly, if we expand around the upper fixed point, all the negative coefficients will be zero. In either case, is the chosen fixed point.

Now from the equation above, we have . But by definition, this is equal to .

By equating the terms of the resulting Laplace series, we get the equation . 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.
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#2
(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 . 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 are the coefficients of the inverse Schröder powerseries.
Incidentally Dmitrii and I just finished an article about exactly that topic, which I append.


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#3
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!
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