[stuck] On the functional equation of the slog : slog(e^z) = slog(z)+1 - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: [stuck] On the functional equation of the slog : slog(e^z) = slog(z)+1 (/showthread.php?tid=853) [stuck] On the functional equation of the slog : slog(e^z) = slog(z)+1 - tommy1729 - 04/28/2014 On the functional equation of the slog : slog(e^z) = slog(z)+1. Related : http://math.eretrandre.org/tetrationforum/showthread.php?tid=850&page=2 ( post 15 mainly ) The issue is here that the functional equation cannot hold everywhere , see also the secondary fixpoints for example ( the related link above ). Also we want to avoid singularities. SO ultimately (imho) 2 questions : 1) when is a region A analytic ? 2) when does the functional equation hold ? And the point of this thread is mainly that the 2 question are related ! The key is the n th derivative : slog(exp(z)) = slog(z) + 1 DERIVATE slog'(exp(z)) exp(z) = slog'(z) 2nd derivative slog''(exp(z)) exp(z) + slog'(exp(z)) = slog''(z) [stuck] Sorry but this idea is a bit stuck. Its complicated. I will update when possible. Feel free to comment. regards tommy1729 RE: [stuck] On the functional equation of the slog : slog(e^z) = slog(z)+1 - tommy1729 - 04/28/2014 Big question : what functional equations for slog(z) do we have in the branches near to the fundamental ?? regards tommy1729