(09/21/2009, 08:06 PM)mike3 Wrote: though I'm not entirely sure how it could be used in constructing the solution to general tetration (any base, incl \( 0 < b \le e^{-e} \)) via the sum equation.
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?
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?
For holomorphic functions satisfying certain conditions described on page 224 we have the summation formula:
\( \sum_{\zeta=c}^z \phi(\zeta) = \frac{1}{2\pi i} \int_C f(z+\zeta) \left(\frac{\pi}{\sin(\pi \zeta)}\right)^2 d\zeta \) where \( f(z)=\int_c^z \phi(\zeta) d\zeta \) and \( C \) is a certain infinite path described on page 225.
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.