12/30/2008, 12:28 PM
Kouznetsov Wrote:bo198214 Wrote:Didnt you say that the series does not converge? Or not to the values of slog?Not to the values of slog. The radius of convergence of the sub-seriesof terms with integer poers of
seems to be
.
Oh I just see, the construction I made above is the regular slog. I.e. the function Knesers starts with. Thatswhy it doesnt yield the values of the kslog, especially it is not real at the real axis. It is the powerseries development of the regular Abel function, while Kneser works with the limit formula of the regular Abel/Schroeder function.
So the continuation of this series to the upper half plane has a singularity at
Lets call this super logarithm rslog while we call your super logarithm kslog. Then we know already that
or
where
By the above consideration we can develop
And every 1-periodic holomorphic function can be developed as:
Then
Now the exponential of a powerseries is again a powerseries:
inserted:
thatswhy we can finally write:
which is slightly more unspecific than Knesers's formula.