09/22/2009, 05:27 AM
(09/22/2009, 01:02 AM)bo198214 Wrote: Not sure, what you mean. I thought this was the open question how to do the summation, its described there. Which sum equation do you mean?
The one for tetration that I mentioned at the beginning of this thread.
(09/22/2009, 01:02 AM)bo198214 Wrote: Unfortunately however it seems that the theory (existence, uniqueness theorems) is well-developed only for at most exponentially growing functions. This seems to be a magical barrier. And perhaps above we lose the uniqueness?
Technically, the continuous sum is not unique regardless of the growth rate, as you can always add a 1-periodic function to any continuous sum and get another function that satisfies the same functional equations. The question then I guess would be to find the "best" sum, perhaps which one could be thought of as an extension of, say, the Bernoulli formula to where it doesn't ordinarily work. Sort of like finding the "best" tetration, but with sums, which seem like an "easier" operation to handle (and yet that is related to tetration via the sum formula, so results from one can be potentially transferred to the other.).
(09/22/2009, 01:02 AM)bo198214 Wrote: However these conditions are not satisfied for e.g. \( e^x \) and there is a certain extension of the method to also handle those cases, but it seems as if clarity is there only about the functions with at most exponential growth.
And tetration displays a variety of behaviors and when it grows it grows up faster than any exponential plus it is not entire.....