05/26/2011, 03:18 AM
(05/20/2011, 06:43 PM)andydude Wrote: That is a really interesting question. First of all, that is one of the bases for which the series expansion of (\exp_b^t(x)) in terms of x is relatively simple with "nice" coefficients, but substituting x=1 (which I think you are talking about) gives a function of t which is not strictly a power series, which makes finding that power series more difficult. Anyways, I believe I have done this before, but I don't have access to my notes right, now, so let me get back to you later today...
Andrew Robbins
Your close to what I'm talking about. I'll write it out in full math format
\(
f(t) = a\, \{t\}\, b =
\left\{
\begin{array}{c l}
\exp_\eta^{\alpha t}(\exp_\eta^{\alpha-t}(a) + \exp_\eta^{\alpha -t}(b)) & t \in [-1,1]\\
\exp_\eta^{\alpha t-1}(b * \exp_\eta^{\alpha 1-t}(a)) & t \in [1,2]\\
\end{array}
\right.
\)
which maps the growth of addition to multiplication to exponentiation. I hope that base eta will behave better than base 2 and base e behaved.
