03/18/2021, 12:55 PM
Unfortunately I think it is not analytic afterall.
It seems the period of A_k(s) is 2 pi i / k.
Hence in the limit we have the period " 0 i " which implies for Re(s) > 1 A(s) = A(Re(s)) = Re(A(s)) and that is clearly not analytic.
This phenomenon follows from the periodicity of exp( - k s ) : 2 pi i / k.
hmm
Maybe if we use another auxiliary function other than exp(- k s) that behaves similar on the real line ? Is that even possible ?
Or maybe we should try a much slower changing function. I considered the uncompleted gamma function too ( although that is faster).
Im even starting to wonder if methods based on composition only excluding riemann mappings has hope for analyticity at all.
So im looking for another auxiliary function f(k,s).
I was considering rational functions and giving up the entire constraint because I dont want too many copies ( like periods or so ).
Im doubting.
regards
tommy1729
It seems the period of A_k(s) is 2 pi i / k.
Hence in the limit we have the period " 0 i " which implies for Re(s) > 1 A(s) = A(Re(s)) = Re(A(s)) and that is clearly not analytic.
This phenomenon follows from the periodicity of exp( - k s ) : 2 pi i / k.
hmm
Maybe if we use another auxiliary function other than exp(- k s) that behaves similar on the real line ? Is that even possible ?
Or maybe we should try a much slower changing function. I considered the uncompleted gamma function too ( although that is faster).
Im even starting to wonder if methods based on composition only excluding riemann mappings has hope for analyticity at all.
So im looking for another auxiliary function f(k,s).
I was considering rational functions and giving up the entire constraint because I dont want too many copies ( like periods or so ).
Im doubting.
regards
tommy1729
