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Principal Branch of the Super-logarithm
Well, it's not as simple as saying, "Here are some cuts". There are a few rules you'll have to follow:

1. Branch cuts must be between pairs of singularities, or between a singularity and infinity, or from infinity to infinity. This is the simple rule we all know and love.
2. The branch cuts cannot pass through an impassable boundary. At most, they can go through an endpoint. Branch cut systems B and C violate this rule.

Thus, only system A would even qualify as a valid system. In that sense, it works fine.

My preference is to draw branch cuts between every "proper" pair of singularities. A "proper" pair of singularities will be logarithmicized/exponentiated images of the line between the primary pair of singularities, or the line between the upper primary singularity and the 2*pi*I offset of the lower primary singularity. This will get all the branch cuts necessary when going into successively "deeper" logarithmic branches. There will be distinct sheets that will be surround on all sides by such branch cuts, or by infinity. The "backbone" is the most basic, with branch cuts to the right, along the imaginary line with real part 0.31813, and infinity to the left, top, and bottom. Within the first layer of logarithmic branches, you'll have a branch cut on the left on an imaginary line segment with real part 0.31813, then curves along the top and bottom that connect at a point at infinity. The next layer in gets more complicated, but in essense the principle is the same.

However, exponential branches will require branch cuts similar to your System A. They get more complicated once you've wrapped around a singularity, because of the weird singularities at the integer tetrates of the base (e.g., 0, 1, e, e^e, etc.), or their logarithmicized/exponentiated images, as the case may be.

I'll draw pictures of what I mean when I get a chance.
~ Jay Daniel Fox

Messages In This Thread
RE: Principal Branch of the Super-logarithm - by jaydfox - 12/11/2007, 03:00 AM

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