I can not yet follow you:

.

edit: oh now I see: you call the fixed point ! Please avoid, is reserved for functions. Kneser calls it , in this thread I keep with his

convention.

edit: well now I also see that you mean the inverted SchrÃ¶der function. As we can derive from the previous formula we get:

No, this still is not your formula, I guess you mean the inverse of the Abel function

this would have the formula:

which is finally your formula

The Abel function maps to some region .

But our final Abel function shall map into the upper halfplane, correspondingly also to and to . Thatswhy we consider the Riemann mapping that biholomorphically maps the union of the by integer translated to the upper halfplane. (though is this marginally different from Kneser's original approach I think it is better accessible.)

Then is the wanted Abel function (slog), and we obtain sexp as:

Not sure what contour exactly you mean, but I think we are close already.

Your theta should be the Riemann mapping that maps the upper halfplane to the union of the by integer translated images .

Please continue from here with unified denotation.

(01/23/2010, 01:01 PM)sheldonison Wrote: , at least I think that is what is meant by .I think the formula is corrupt, the correct formula should be:

.

edit: oh now I see: you call the fixed point ! Please avoid, is reserved for functions. Kneser calls it , in this thread I keep with his

convention.

edit: well now I also see that you mean the inverted SchrÃ¶der function. As we can derive from the previous formula we get:

No, this still is not your formula, I guess you mean the inverse of the Abel function

this would have the formula:

which is finally your formula

Quote:Is the real valued sexp at the real axis connected to the Schroeder equation, chi, by a complex 1-cyclic function,

The Abel function maps to some region .

But our final Abel function shall map into the upper halfplane, correspondingly also to and to . Thatswhy we consider the Riemann mapping that biholomorphically maps the union of the by integer translated to the upper halfplane. (though is this marginally different from Kneser's original approach I think it is better accessible.)

Then is the wanted Abel function (slog), and we obtain sexp as:

Quote:Where is the Riemann mapping of contour of the limit of iterating the natural logarithm, starting with the real interval, 0..1 [color=#0000CD](which corresponds to sexp(z) from z=0 to z=1).

Not sure what contour exactly you mean, but I think we are close already.

Your theta should be the Riemann mapping that maps the upper halfplane to the union of the by integer translated images .

Please continue from here with unified denotation.