I have been playing around with the complex equivalent of a piecewise-defined slog, and with the sections of it that should appear on its 0th branch. I feel like what Jay calls the "backbone" along with power series along the real axis, should be in the 0th branch. Since the real part of the real axis is generally greater than the backbone, the analytic continuation from the positive real axis towards the backbone (going through a line with imaginary part approximately pi) should bring us to the 1st branch. An analytic continuation from the backbone (going through a line with imaginary part approximately pi) should then bring us to the -1st branch. Where the branch cuts are is really arbitrary, and is anyone's guess as to where would be best. This is what I'm writing this post about, because I don't know which branch cut is nicest.
In the images below, the big dots in the the branch-cut diagram are the fixed points of e^x. The smaller dots are the 2 pi shifting of the fixed points, which are always a singularity on some branch. The thick lines are the branch cuts.
Branch cut system A (branch cuts, real part, imaginary part)
seems to be the best of all of these, but does not preserve periodicity. It has the fewest branch cuts, and does not avoid the second pair of fixed points \( \{-W_{-2}(-1),-W_1(-1)\} \) which the others do avoid. All consider the first pair of fixed points, and the singularities derived from them. If the second pair of fixed points do become a problem (I haven't tried extending the domain that far yet) then we can just make branch cuts from that pair, away from the real line).
Branch cut system B (branch cuts, real part, imaginary part)
has the nicest real-part graph, and preserves periodicity across the imaginary axis. All branch systems should have a periodic backbone, but where the periodicity ends is a matter of taste, I suppose.
Branch cut system C (branch cuts, real part, imaginary part)
was actually my first attempt, because this is how Jay describes the rotation around the first pair of fixed points. The thing I like about this one is that it contains both real functions. The real function along the real axis, and the real function along the (real + ipi) axis, which has real output since:
\( \text{slog}(x+\pi i) = \text{slog}(e^{x+\pi i}) - 1 = \text{slog}(-e^{x}) - 1 \)
I'm sorry I haven't taken the time to convert these images to png, but knowing you guys, you'd probably ask for pdf anyways...
Andrew Robbins
In the images below, the big dots in the the branch-cut diagram are the fixed points of e^x. The smaller dots are the 2 pi shifting of the fixed points, which are always a singularity on some branch. The thick lines are the branch cuts.
Branch cut system A (branch cuts, real part, imaginary part)
seems to be the best of all of these, but does not preserve periodicity. It has the fewest branch cuts, and does not avoid the second pair of fixed points \( \{-W_{-2}(-1),-W_1(-1)\} \) which the others do avoid. All consider the first pair of fixed points, and the singularities derived from them. If the second pair of fixed points do become a problem (I haven't tried extending the domain that far yet) then we can just make branch cuts from that pair, away from the real line).
Branch cut system B (branch cuts, real part, imaginary part)
has the nicest real-part graph, and preserves periodicity across the imaginary axis. All branch systems should have a periodic backbone, but where the periodicity ends is a matter of taste, I suppose.
Branch cut system C (branch cuts, real part, imaginary part)
was actually my first attempt, because this is how Jay describes the rotation around the first pair of fixed points. The thing I like about this one is that it contains both real functions. The real function along the real axis, and the real function along the (real + ipi) axis, which has real output since:
\( \text{slog}(x+\pi i) = \text{slog}(e^{x+\pi i}) - 1 = \text{slog}(-e^{x}) - 1 \)
I'm sorry I haven't taken the time to convert these images to png, but knowing you guys, you'd probably ask for pdf anyways...
Andrew Robbins
