I'm (morbidly?) fascinated by iterated exponentials and I like the idea of extrapolating a continuous extension of a function from simple behavior at a limit. The stuff I was doing with sinh and tanh, seemed a bit arbitrary, though. exp() and ln() behave simply near their fixed points. Say this one: ln( z0 ) = z0 = exp( z0 ) ~= 0.318 + 1.337 i Forgive my using the letter d in the following: lim d -> 0 exp( z0 + d ) = z0 + z0 d and so lim d -> 0 exp^n( z0 + d ) = z0 + z0^n d The current place I've wandered to is to define lim n -> oo iter_exp( z ) = exp^n( z0 + z0^(z-n) ) That gives a sort of analogy to e: iter_exp( 0 ) = lim n -> oo exp^n( z0 + z0^-n ) ~= 1.149 + 1.229 i Similar to how exp( 1 ) = lim n -> oo ( 1 + 1/n )^n It looks like a faulty analogy; I haven't come up with a better one so far. The closest thing on Wikipedia is http://en.wikipedia.org/wiki/Tetration#Extension_to_complex_heights which looks very different but I just glanced. Before the picture, other bits: iter_exp( -oo ) = z0 iter_exp( +oo i ) = z0 (Half the plane maps into a disk-like area around the fixed point.) iter_exp( z + 1 ) = exp( iter_exp( z ) ) Which is nice, but there are no z's where iter_exp(z) = 0 or 1 or e or e^e... I may have to learn some complex analysis. Or maybe category theory or LaTeX. Here are pics of a map of the thing before it gets branchy: Two views, each a 75K jpg: http://www.mac-guyver.com/switham/2010/03/Superbola/ln_fixed_point.jpg http://www.mac-guyver.com/switham/2010/03/Superbola/ln_fixed_brain.jpg 395K pdf of the first view: http://www.mac-guyver.com/switham/2010/03/Superbola/ln_fixed_point.pdf Python code: http://www.mac-guyver.com/switham/2010/03/Superbola/ln_fixed_point.py uses this: http://www.mac-guyver.com/switham/2010/03/Superbola/tinyplot.py --Steve