11/05/2009, 02:12 PM

Hey folks,

until now we mainly considered holomorphy in the second argument of tetration:

.

If we now fix and demand also the function to be analytic on one small interval , the function is already determined for all other bases, particularly for bases by analytic continuation along the real axis (if there is no singularity at ).

For bases 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 continues for . I guess it has a branch point at and continues to complex values beyond .

I will explore this thought more concretely in the following posts.

until now we mainly considered holomorphy in the second argument of tetration:

.

If we now fix and demand also the function to be analytic on one small interval , the function is already determined for all other bases, particularly for bases by analytic continuation along the real axis (if there is no singularity at ).

For bases 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 continues for . I guess it has a branch point at and continues to complex values beyond .

I will explore this thought more concretely in the following posts.