11/05/2009, 02:12 PM
Hey folks,
until now we mainly considered holomorphy in the second argument \( t \) of tetration:
\( b [4] t = {\exp_b}^{\circ t}(1) \).
If we now fix \( t \) and demand also the function \( f_t(b)=b [4] t \) to be analytic on one small interval \( b\in(b_1,b_2) \), the function \( f_t \) is already determined for all other bases, particularly for bases \( b>e^{1/e} \) by analytic continuation along the real axis (if there is no singularity at \( b=e^{1/e} \)).
For bases \( 1<b<e^{1/e} \) we know that the regular iteration at the lower fixed point is analytic there and continuable to 1.
So the question would be to what values the regular iteration \( f_t \) continues for \( b>e^{1/e} \). I guess it has a branch point at \( b^{1/b} \) and continues to complex values beyond \( e^{1/e} \).
I will explore this thought more concretely in the following posts.
until now we mainly considered holomorphy in the second argument \( t \) of tetration:
\( b [4] t = {\exp_b}^{\circ t}(1) \).
If we now fix \( t \) and demand also the function \( f_t(b)=b [4] t \) to be analytic on one small interval \( b\in(b_1,b_2) \), the function \( f_t \) is already determined for all other bases, particularly for bases \( b>e^{1/e} \) by analytic continuation along the real axis (if there is no singularity at \( b=e^{1/e} \)).
For bases \( 1<b<e^{1/e} \) we know that the regular iteration at the lower fixed point is analytic there and continuable to 1.
So the question would be to what values the regular iteration \( f_t \) continues for \( b>e^{1/e} \). I guess it has a branch point at \( b^{1/b} \) and continues to complex values beyond \( e^{1/e} \).
I will explore this thought more concretely in the following posts.
