[stuck] On the functional equation of the slog : slog(e^z) = slog(z)+1

[tommy1729]

On the functional equation of the slog : slog(e^z) = slog(z)+1.

Related : http://math.eretrandre.org/tetrationforu...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

[tommy1729]

Big question : what functional equations for slog(z) do we have in the branches near to the fundamental ??

regards

tommy1729