11/12/2009, 03:23 PM
actually I think that the regular tetration at the lower fixed point is not continuable to bases \( >e^{1/e} \).
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 \( \exp_b \) at the fixed point.
2. If \( a \) is the fixed point then the derivation is \( \log(a) \)
3. The fixed point is given by \( a = \exp(-W(-\log(b))) \), or \( \log(a)=-W(-\log(b)) \).
4. The problematic value of the derivation at the fixed point is \( |\log(a)|=1 \). In this case \( \exp_b^{\circ t} \) is not analytic at \( a \). I guess this implies that \( b\mapsto\exp_b^{\circ t}(1) \) is not analytic at \( b \).
5. \( |\log(a)|=1 \) is exactly at those \( b \) in the image of the unit circle under of the inverse of \( b\mapsto -W(-\log(b)) \), i.e. under \( z\mapsto\exp(z \exp(-z)) \) 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 \( \exp_b \) at the fixed point.
2. If \( a \) is the fixed point then the derivation is \( \log(a) \)
3. The fixed point is given by \( a = \exp(-W(-\log(b))) \), or \( \log(a)=-W(-\log(b)) \).
4. The problematic value of the derivation at the fixed point is \( |\log(a)|=1 \). In this case \( \exp_b^{\circ t} \) is not analytic at \( a \). I guess this implies that \( b\mapsto\exp_b^{\circ t}(1) \) is not analytic at \( b \).
5. \( |\log(a)|=1 \) is exactly at those \( b \) in the image of the unit circle under of the inverse of \( b\mapsto -W(-\log(b)) \), i.e. under \( z\mapsto\exp(z \exp(-z)) \) which is just the boundary of the Shell-Thron region.
