11/22/2008, 04:22 PM
It seems we dont speak about the same thing.
So my first question is:
\( \sqrt{\exp}(L_{e,1})=L_{e,1} \)?
where \( L_{e,1}\approx 2.062 + 7.589i \) is the fixed point of \( \exp \) on the upper halfplane that has the second lowest distance to the real line.
I mean one could expect that a half iterate has the same fixed points as the function itself. Unfortunately I can not verify the above question from the picture.
So if it turns out that \( \sqrt{\exp}(L_{e,1})\neq L_{e,1} \) on the given domain of definition, then I would expect that there is some branch\( \sqrt{\exp}_k \) such that \( \sqrt{\exp}_k(L_{e,1})=L_{e,1} \). And further if there is such a branch (made up at the branch point \( L \)) then I would expect that \( \sqrt{\exp}_k \) has a (maybe isolated) singularity at \( L_{e,1} \).
So my first question is:
\( \sqrt{\exp}(L_{e,1})=L_{e,1} \)?
where \( L_{e,1}\approx 2.062 + 7.589i \) is the fixed point of \( \exp \) on the upper halfplane that has the second lowest distance to the real line.
I mean one could expect that a half iterate has the same fixed points as the function itself. Unfortunately I can not verify the above question from the picture.
So if it turns out that \( \sqrt{\exp}(L_{e,1})\neq L_{e,1} \) on the given domain of definition, then I would expect that there is some branch\( \sqrt{\exp}_k \) such that \( \sqrt{\exp}_k(L_{e,1})=L_{e,1} \). And further if there is such a branch (made up at the branch point \( L \)) then I would expect that \( \sqrt{\exp}_k \) has a (maybe isolated) singularity at \( L_{e,1} \).