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Unique Holomorphic Super Logarithm
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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 that has a convergence radius of at least when developed at 0 and that maps an open set containing - which is defined below - (or ) biholomorphically to some set , that contains for each real value a horizontal line of length with imaginary part .

Here is the first fixed point of and can be roughly seen on the following picture:
   

The idea behind is the following. If we look at the straight line between and which can be given by for , then the area bounded by and can be considered as an initial area from which you can derive for example the values on or on by , or .

You can see very well on the picture that lies on the circle with radius (red dashed line). This can be easily derived:
By we know that and hence
which is an arc with radius |L| around 0.

To be more precise we define exactly what we mean:
Let be the set enclosed by and for , included but excluded. This set is not open and hence not a domain, but if we move slightly to the left we get a domain containing .

We call a function defined on the domain a super exponential iff it satisfies and for all such that .

We call a function defined on the domain a super logarithm iff for each (but not necessarily for each ).

With those specifications we can come to the proof.
Proof. Assume there are two holomorphic super logarithms and .
Then both are defined on the domain and map it bihomorphically, say and . is holomorphic on the domain . By the condition on and by it can be continued to an entire function. The same is true for , which is the inverse of and hence must as it was shown in the proof in universal uniqueness criterion II.
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Unique Holomorphic Super Logarithm - by bo198214 - 11/18/2008, 06:18 PM

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