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Kneser's Super Logarithm
#11
I can not yet follow you:
(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 Schröder 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 Smile

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.
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Messages In This Thread
Kneser's Super Logarithm - by bo198214 - 11/19/2008, 02:20 PM
RE: Kneser's Super Logarithm - by bo198214 - 11/19/2008, 03:25 PM
RE: Kneser's Super Logarithm - by sheldonison - 01/23/2010, 01:01 PM
RE: Kneser's Super Logarithm - by mike3 - 01/25/2010, 06:35 AM
RE: Kneser's Super Logarithm - by sheldonison - 01/25/2010, 07:42 AM
RE: Kneser's Super Logarithm - by mike3 - 01/26/2010, 06:24 AM
RE: Kneser's Super Logarithm - by sheldonison - 01/26/2010, 01:22 PM
RE: Kneser's Super Logarithm - by mike3 - 01/27/2010, 06:28 PM
RE: Kneser's Super Logarithm - by sheldonison - 01/27/2010, 08:30 PM
RE: Kneser's Super Logarithm - by mike3 - 01/28/2010, 08:52 PM
RE: Kneser's Super Logarithm - by sheldonison - 01/28/2010, 10:08 PM
RE: Kneser's Super Logarithm - by mike3 - 01/29/2010, 06:43 AM
RE: Kneser's Super Logarithm - by bo198214 - 01/26/2010, 11:19 PM
RE: Kneser's Super Logarithm - by sheldonison - 01/27/2010, 07:51 PM
RE: Kneser's Super Logarithm - by bo198214 - 11/22/2008, 06:11 PM
RE: Kneser's Super Logarithm - by bo198214 - 11/23/2008, 01:00 PM

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