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+--- Thread: properties of abel functions in general (/showthread.php?tid=371)
properties of abel functions in general - Base-Acid Tetration - 10/24/2009
Let us summarize what are known about superfunctions, abel functions, etc.
*Let f be a holo. function. Let A be an abel function of f. if a is a fixed point of f, then A has a logarithmic branch point at A. not necessarily a log branch pt. but still some kind of singularity.
RE: properties of abel functions in general - andydude - 10/24/2009
(10/24/2009, 12:28 AM)Base-Acid Tetration Wrote: *Let f be a holo. function. Let A be an abel function of f. if a is a fixed point of f, then A has a logarithmic branch point at A.
How do you know that? It might be a simple pole or something else... It will definitely be a singularity/undefined, but I'm not convinced that it will be any particular kind of singularity/branchpoint... I would need more proof.
RE: properties of abel functions in general - Base-Acid Tetration - 10/24/2009
I'm thinking that IF the abel function of f as a simple pole at L (fixed point), the f-iterational (superfunction of f) must decay to L as |z| -> infinity (no matter what the argument of z is), and that f(z) =/= L any z =/= complex infinity.
for example, 1/z, which has a simple pole at 0, is an abel function (also the iterational/superfunction) of z/(z+1), and z/(z+1) has a fixed point at zero. 1/z, being its own inverse, also decays to zero asymptotically as |z| -> infinity.
more complicated examples have the same pattern. z^-n's base function is \( \lbrace \operatorname{sgn}_{(1,2,...n)} \rbrace \)(z^n/(z^n-1))^1/n. the "sgn" thing is the symbol I invented for the nth roots of unity. (it's like the plus-minus sign.)
the inverse of z^-n is z^-1/n which has n branches, the k-th branch of which corresponds to the "k-th side" of z^-n's pole of order n. (at the k-th branch, where |z| is large is mapped to a "wedge" (which "points" to the pole at 0) whose angular measure is 2k*pi/n.)
RE: properties of abel functions in general - bo198214 - 10/24/2009
(10/24/2009, 12:28 AM)Base-Acid Tetration Wrote: Let us summarize what are known about superfunctions, abel functions, etc.
*Let f be a holo. function. Let A be an abel function of f. if a is a fixed point of f, then A has a logarithmic branch point at A. not necessarily a log branch pt. but still some kind of singularity.
If you consider regular iteration at a hyperbolic fixed point \( z_0 \), then definitely the Abel function has a logarithmic singularity there. It is of the form:
\( \log_c(z-z_0)+p(z) \) where \( c=f'(z_0) \) and \( p \) is some analytic function in the vicinity of \( z_0 \).