Oh, wow! I couldn't get a direct solution for the integration of atanh incorporating complex numbers, but by using the Taylor series for atanh, so far I have been able to find solutions to the first and second order where the complex numbers just fall out each time. The first order doesn't involve elliptics or anything complicated and I'm thinking (hoping) the other orders might not either. I may well be on my way to a solution for a uniformly dense disk after all. The result would still be in the form of an infinite series, though, unless I can find the way the pattern for it runs and use another function for that as well. I'll keep working on it.

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Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."