06/01/2009, 06:14 PM

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.

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.