The idea is as follows.
all branches together should have the range of the full complex plane.
In fact all branches together should map bijectively to the full complex plane.
If there exist a jordan curve with infinite length (lenght on the complex plane) on the riemann sphere such that every point x on it belongs to the range of the main branch and such that x + k i for any k with 0<k<2pi also belongs to the main branch then it is unavoidable that the branch above the main branch is exp + 2 pi i.
Notice that exp + 2pi i also has 2 fixpoints and the same singularities as exp.
SO it might be possible.
Al this needs to be made formal of course.
regards
tommy1729
all branches together should have the range of the full complex plane.
In fact all branches together should map bijectively to the full complex plane.
If there exist a jordan curve with infinite length (lenght on the complex plane) on the riemann sphere such that every point x on it belongs to the range of the main branch and such that x + k i for any k with 0<k<2pi also belongs to the main branch then it is unavoidable that the branch above the main branch is exp + 2 pi i.
Notice that exp + 2pi i also has 2 fixpoints and the same singularities as exp.
SO it might be possible.
Al this needs to be made formal of course.
regards
tommy1729