------ Original Message ------ Received: Mon, 24 May 2010 03:59:20 PM PDT From: Steve Witham <sw@tiac.net> To: math-fun@mailman.xmission.com Subject: [math-fun] iter_exp()

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.

... The current place I've wandered to is to define

lim n -> oo iter_exp( z ) = exp^n( z0 + z0^(z-n) )

...

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.

I don't think there are z's that map to 0, but remember exp is not 1-1 in the complex plane. Find a place where iter_exp(z) = 2 pi i and that yields a place where iter_exp(z) = 1 I find one around 1.511 - .217i Giving us 1 at around 2.511 - .217i Christian

...

--Steve