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Structure of the graph of the tetration function
#1
I saw the graph of an extension of the tetrational function (to the base e) on the complex numbers here:

http://upload.wikimedia.org/wikipedia/co...pFig1d.jpg

But what is going on there near the real axis? It seems to be getting very convoluted in quite intriguing ways, and the computer appears to have been unable to plot it all there given the missing areas of graph, presumably due to huge values popping up somewhere in the computation, too big for it to handle. No "special functions" I'm aware of have this interesting behavior, so if tetration was eventually made into one it would make quite an interesting addition.
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