(04/20/2010, 02:48 AM)mike3 Wrote:

Really really interesting.

Quote:with the term at interpreted as , so,

.

However, when viewed in the complex plane, we see that exp-series are just Fourier series

.

which represent a periodic function with period .

Thus it would seem that only periodic functions can be continuum-summed this way. Tetration is not periodic, so how could this help?

This is not completely true. The regular tetration in the base range has the form

where is a holomorphic function with and (this is the inverse of the Schröder function), is the fixed point and (and s is some arbitrary constant which you would choose to ascertain that .

So it is periodic.

If I consider Kneser's approach for we can obtain a related representation.

Kneser's approach also starts with the regular iteration at a complex fixed point. Lets call the corresponding superfunction which is not real on the real axis. If we call the real superfunction then we get that is a 1-periodic function, i.e.

This is not periodic, nonetheless we can expand it similar to a periodic function.

(for simplicity consdier contained in ).

The last factor is a 1-periodic function:

If we insert this into our previous equation:

So we have a double exponential series instead of a single series, but nevertheless you again can apply your exponential summation. Though I in the moment have not the time to carry it out myself (so either you do it or I do it later).