11/18/2008, 06:18 PM
When I continued to find a suitable domain of definition for the slog such that the it is unique there by the universal uniqueness criterion II, I followed a line of thoughts I will describe later and came up with the following condition:
Proposition. There is at most one holomorphic super logarithm \( \text{slog} \) that has a convergence radius of at least \( |L| \) when developed at 0 and that maps an open set \( G^o \) containing \( G \) - which is defined below - (or \( \log(G) \)) biholomorphically to some set \( T \), that contains for each real value \( y \) a horizontal line of length \( >1 \) with imaginary part \( y \).
Here \( L \) is the first fixed point of \( \exp \) and \( G \) can be roughly seen on the following picture:
The idea behind is the following. If we look at the straight line between \( L \) and \( L^\ast \) which can be given by \( \el(t)=\Re(L)+i\Im(L)t \) for \( -1<t<1 \), then the area \( G \) bounded by \( \el \) and \( e^{\el} \) can be considered as an initial area from which you can derive for example the values on \( e^G \) or on \( \log(G) \) by \( \text{slog}(\exp(z))=\text{slog}(z)+1 \), or \( \text{slog}(\log(z))=\text{slog}(z)-1 \).
You can see very well on the picture that \( e^{\el} \) lies on the circle with radius \( |L| \) (red dashed line). This can be easily derived:
By \( e^L=L \) we know that \( e^{\Re(L)}=|e^L|=|L| \) and hence
\( e^{\el(t)}=e^{\Re(L)+i\Im(L)t}=|L|e^{i\Im(L)t} \) which is an arc with radius |L| around 0.
To be more precise we define exactly what we mean:
Let \( G \) be the set enclosed by \( \el(t) \) and \( e^{\el(t)} \) for \( -1<t<1 \), \( \el \) included but \( e^\el \) excluded. This set is not open and hence not a domain, but if we move \( \el \) slightly to the left we get a domain \( G^o \) containing \( G \).
We call a function \( \text{sexp} \) defined on the domain \( D \) a super exponential iff it satisfies \( \text{sexp}(0)=1 \) and \( \text{sexp}(z+1)=\exp(\text{sexp}(z)) \) for all \( z \) such that \( z, z+1\in D \).
We call a function \( \text{slog} \) defined on the domain \( H \) a super logarithm iff \( \text{sexp}(\text{slog}(z))=z \) for each \( z\in H \) (but not necessarily \( \text{slog}(\text{sexp}(z)) \) for each \( z\in\text{slog}(H) \)).
With those specifications we can come to the proof.
Proof. Assume there are two holomorphic super logarithms \( f^{-1}: G_1^o\leftrightarrow T_1 \) and \( g^{-1}: G_2^o\leftrightarrow T_2 \) .
Then both are defined on the domain \( G^o=G_1^o\cap G_2^o \) and map it bihomorphically, say \( f^{-1}:G^o\leftrightarrow T_1' \) and \( g^{-1}:G^o\leftrightarrow T_2' \). \( \delta:=g^{-1}\circ f \) is holomorphic on the domain \( T_1' \). By the condition on \( T_1 \) and by \( \delta(z+1)=\delta(z)+1 \) it can be continued to an entire function. The same is true for \( \delta_2:=f^{-1}\circ g \), which is the inverse of \( \delta \) and hence must \( \delta(z)=z \) as it was shown in the proof in universal uniqueness criterion II.\( \boxdot \)
Proposition. There is at most one holomorphic super logarithm \( \text{slog} \) that has a convergence radius of at least \( |L| \) when developed at 0 and that maps an open set \( G^o \) containing \( G \) - which is defined below - (or \( \log(G) \)) biholomorphically to some set \( T \), that contains for each real value \( y \) a horizontal line of length \( >1 \) with imaginary part \( y \).
Here \( L \) is the first fixed point of \( \exp \) and \( G \) can be roughly seen on the following picture:
The idea behind is the following. If we look at the straight line between \( L \) and \( L^\ast \) which can be given by \( \el(t)=\Re(L)+i\Im(L)t \) for \( -1<t<1 \), then the area \( G \) bounded by \( \el \) and \( e^{\el} \) can be considered as an initial area from which you can derive for example the values on \( e^G \) or on \( \log(G) \) by \( \text{slog}(\exp(z))=\text{slog}(z)+1 \), or \( \text{slog}(\log(z))=\text{slog}(z)-1 \).
You can see very well on the picture that \( e^{\el} \) lies on the circle with radius \( |L| \) (red dashed line). This can be easily derived:
By \( e^L=L \) we know that \( e^{\Re(L)}=|e^L|=|L| \) and hence
\( e^{\el(t)}=e^{\Re(L)+i\Im(L)t}=|L|e^{i\Im(L)t} \) which is an arc with radius |L| around 0.
To be more precise we define exactly what we mean:
Let \( G \) be the set enclosed by \( \el(t) \) and \( e^{\el(t)} \) for \( -1<t<1 \), \( \el \) included but \( e^\el \) excluded. This set is not open and hence not a domain, but if we move \( \el \) slightly to the left we get a domain \( G^o \) containing \( G \).
We call a function \( \text{sexp} \) defined on the domain \( D \) a super exponential iff it satisfies \( \text{sexp}(0)=1 \) and \( \text{sexp}(z+1)=\exp(\text{sexp}(z)) \) for all \( z \) such that \( z, z+1\in D \).
We call a function \( \text{slog} \) defined on the domain \( H \) a super logarithm iff \( \text{sexp}(\text{slog}(z))=z \) for each \( z\in H \) (but not necessarily \( \text{slog}(\text{sexp}(z)) \) for each \( z\in\text{slog}(H) \)).
With those specifications we can come to the proof.
Proof. Assume there are two holomorphic super logarithms \( f^{-1}: G_1^o\leftrightarrow T_1 \) and \( g^{-1}: G_2^o\leftrightarrow T_2 \) .
Then both are defined on the domain \( G^o=G_1^o\cap G_2^o \) and map it bihomorphically, say \( f^{-1}:G^o\leftrightarrow T_1' \) and \( g^{-1}:G^o\leftrightarrow T_2' \). \( \delta:=g^{-1}\circ f \) is holomorphic on the domain \( T_1' \). By the condition on \( T_1 \) and by \( \delta(z+1)=\delta(z)+1 \) it can be continued to an entire function. The same is true for \( \delta_2:=f^{-1}\circ g \), which is the inverse of \( \delta \) and hence must \( \delta(z)=z \) as it was shown in the proof in universal uniqueness criterion II.\( \boxdot \)
