So that means \( \{-W_{-2}(-1),-W_1(-1)\} \) are only fixed points of the exponential function, and not fixed-points of the principal branch of the logarithm, since it takes it takes the modulus (modulo 2ipi) of everything so it surrounds the real axis. The Imaginary part of \( Im[-W_{-2}(-1)] = 7.5886 \) which is well beyond the range of the principal branch of the logarithm. Since I am using the principal branch of the logarithm to move points with positive real part into the unit circle, these fixed points would not show up on this branch as singularities. If there was another branch that was using the branch of the logarithm in which these points are fixed points, then of course we would need another branch cut for them, since they would appear as singularities. But as it is, I think branch cut system A is the simplest choice for the principal branch of the base-e super-logarithm.
Andrew Robbins
Andrew Robbins
