06/20/2009, 07:27 PM
(This post was last modified: 07/06/2009, 12:08 AM by Base-Acid Tetration.)

Theorem.

Let be a function that is biholomorphic on both of the initial regions and that share as boundaries the same conjugate pair of fixed points of f. Also let f have no other fixed points. Let there be two Abel functions of f, and , that are biholomorphic on these initial regions and satisfy A(d) = c.

For each f, there exists exactly one biholomorphism on a single simply connected open set such that (i is an index that can be 1 or 2) i.e. there is an analytic continuation, and it's unique.

Proof.

1.

Let be disjoint, simply connected domains that have as boundaries:

(1) , disjoint curves which are homeomorphic to (0,1);

(2) and , which are boundaries of, but not contained in, .

2.

Let be a biholomorphism on , where , (tried to make the domain of biholomorphism into an open set) that:

(1) bijects to ;

(2) has a conjugate pair of fixed points and ;

(3) has no other fixed points in the domain of biholomorphy.

3.

(1) Let , a biholomorphism on , and , a biholomorphism on , both satisfy for all applicable z. (for all z such that A(f(z)) is defined)

(2) Let for some .

to be continued

Let be a function that is biholomorphic on both of the initial regions and that share as boundaries the same conjugate pair of fixed points of f. Also let f have no other fixed points. Let there be two Abel functions of f, and , that are biholomorphic on these initial regions and satisfy A(d) = c.

For each f, there exists exactly one biholomorphism on a single simply connected open set such that (i is an index that can be 1 or 2) i.e. there is an analytic continuation, and it's unique.

Proof.

1.

Let be disjoint, simply connected domains that have as boundaries:

(1) , disjoint curves which are homeomorphic to (0,1);

(2) and , which are boundaries of, but not contained in, .

2.

Let be a biholomorphism on , where , (tried to make the domain of biholomorphism into an open set) that:

(1) bijects to ;

(2) has a conjugate pair of fixed points and ;

(3) has no other fixed points in the domain of biholomorphy.

3.

(1) Let , a biholomorphism on , and , a biholomorphism on , both satisfy for all applicable z. (for all z such that A(f(z)) is defined)

(2) Let for some .

to be continued