Complementing your post with some pictures ...
We have the two primary conjugated fixpoints of b^x, b>eta.
Say \( L^+ \) is the on the upper halfplane and \( L^- \) the one on the lower halfplane.
As you write they are given by:
\( L^+(b)=L_{-1}=\frac{W_{-1}(-\log(b))}{-\log(b)} \)
\( L^-(b)=L_0=\frac{W_{0}(-\log(b))}{-\log(b)} \)
Where \( W_{-1} \) and \( W_0 \) are the standard branches given in computer algebra systems. For these branches we have anticlockwise:
\( L_{-1}(e+0i) \)=secondary upper fp\( \approx 2.06227772959828 + 7.58863117847251*I \)
\( L_{-1}(\sqrt{2})=4 \)
\( L_{-1}(e-0i) \)=primary upper fp\( \approx 0.318131505204764 + 1.33723570143069*I \)
\( L_{0}(e+0i) \)=primary upper fp
\( L_{0}(\sqrt{2})=2 \)
\( L_0(e-0i) \)=primary lower fp\( \approx 0.318131505204764 - 1.33723570143069*I \)
\( L_0 \) has a cut on \( (\eta,\infty) \) and \( L_k \) has a cut on \( (1,\infty) \) for \( k\neq 0 \), because \( W_0 \) has a cut on \( (-1/e,-\infty) \) and all other branches of \( W \) have a cut on \( (0,-\infty) \) where 0 is a singularity.
so we replace \( W_{0} \) in the upper halfplane by \( W_{-1} \) in \( L_0 \) yielding the function \( M^+ \) and replace \( W_{0} \) by \( W_1 \) in the lower halfplane in \( L_0 \) yielding \( M^- \).
Then both have the following color contour plots:
\( M^+(e)= \)upper primary fp
\( M^+(\sqrt{2},+0i)=2 \)
\( M^+(\sqrt{2},-0i)=4 \)
\( M^-(e)= \)lower primary fp
\( M^-(\sqrt{2},+0i)=4 \)
\( M^-(\sqrt{2},-0i)=2 \)
Lets see what happens, with the fixpoints \( M^\pm(b) \) when moving on the circle \( b=\beta(t)=\eta+ (\eta-\sqrt{2})e^{\pi i t} \) for \( -1<t <1 \) in the next post.
We have the two primary conjugated fixpoints of b^x, b>eta.
Say \( L^+ \) is the on the upper halfplane and \( L^- \) the one on the lower halfplane.
As you write they are given by:
\( L^+(b)=L_{-1}=\frac{W_{-1}(-\log(b))}{-\log(b)} \)
\( L^-(b)=L_0=\frac{W_{0}(-\log(b))}{-\log(b)} \)
Where \( W_{-1} \) and \( W_0 \) are the standard branches given in computer algebra systems. For these branches we have anticlockwise:
\( L_{-1}(e+0i) \)=secondary upper fp\( \approx 2.06227772959828 + 7.58863117847251*I \)
\( L_{-1}(\sqrt{2})=4 \)
\( L_{-1}(e-0i) \)=primary upper fp\( \approx 0.318131505204764 + 1.33723570143069*I \)
\( L_{0}(e+0i) \)=primary upper fp
\( L_{0}(\sqrt{2})=2 \)
\( L_0(e-0i) \)=primary lower fp\( \approx 0.318131505204764 - 1.33723570143069*I \)
\( L_0 \) has a cut on \( (\eta,\infty) \) and \( L_k \) has a cut on \( (1,\infty) \) for \( k\neq 0 \), because \( W_0 \) has a cut on \( (-1/e,-\infty) \) and all other branches of \( W \) have a cut on \( (0,-\infty) \) where 0 is a singularity.
so we replace \( W_{0} \) in the upper halfplane by \( W_{-1} \) in \( L_0 \) yielding the function \( M^+ \) and replace \( W_{0} \) by \( W_1 \) in the lower halfplane in \( L_0 \) yielding \( M^- \).
Then both have the following color contour plots:
\( M^+(e)= \)upper primary fp
\( M^+(\sqrt{2},+0i)=2 \)
\( M^+(\sqrt{2},-0i)=4 \)
\( M^-(e)= \)lower primary fp
\( M^-(\sqrt{2},+0i)=4 \)
\( M^-(\sqrt{2},-0i)=2 \)
Lets see what happens, with the fixpoints \( M^\pm(b) \) when moving on the circle \( b=\beta(t)=\eta+ (\eta-\sqrt{2})e^{\pi i t} \) for \( -1<t <1 \) in the next post.