11/12/2009, 03:23 PM
actually I think that the regular tetration at the lower fixed point is not continuable to bases 
.
Actually I think the Shell-Thron region is the domain of holomorphy, every point on the boundary is singular. This became clear to me some days ago:
1. The regular iteration depends on the derivative of
at the fixed point.
2. If
is the fixed point then the derivation is )
3. The fixed point is given by)))
, or =-W(-\log(b)))
.
4. The problematic value of the derivation at the fixed point is|=1)
. In this case 
is not analytic at . I guess this implies that )
is not analytic at 
.
5. is exactly at those in the image of the unit circle under of the inverse of ))
, i.e. under ))
which is just the boundary of the Shell-Thron region.
Actually I think the Shell-Thron region is the domain of holomorphy, every point on the boundary is singular. This became clear to me some days ago:
1. The regular iteration depends on the derivative of
2. If
3. The fixed point is given by
4. The problematic value of the derivation at the fixed point is
5.