So up to now we have a Schroeder function in a vicinity of the fixed point . This function satisfies:

The next thing Kneser does is to analytically continue from this small vicinity to the whole upper halfplane without the points , that is:

.

He first verifies that for each (where the cut of the logarithm is the usual one, i.e. .)

And then he continues the function along the increasing sets , which contain all the points such that is contained in that initial vicinity of .

The inverse function (in a vicinity of ) satisifies:

and hence can be continued to the whole complex plane, i.e. is an entire function.

To see the properties of on he considers the following lines and areas. Each increasing index number indicates the application of exponentiation, for example , . The lines are without end points, the areas are without boundaries.

These areas are mapped by to (the letters indicate the source area):

Now is simply connected and does not contain 0 (only in the boundary). Hence it is possible to define a holomorphic logarithm on that domain, he defines

on . Which then satisfies

.

And this is the image under , considering that is biholomorphic:

Define and define , then we have and . Define . So one can see that the boundary of consists only of the cyan and violet arcs, hence the points of the real axis in are mapped to the boundary of . This property is used in the final step of Kneser's construction, which follows in my next post.

The next thing Kneser does is to analytically continue from this small vicinity to the whole upper halfplane without the points , that is:

.

He first verifies that for each (where the cut of the logarithm is the usual one, i.e. .)

And then he continues the function along the increasing sets , which contain all the points such that is contained in that initial vicinity of .

The inverse function (in a vicinity of ) satisifies:

and hence can be continued to the whole complex plane, i.e. is an entire function.

To see the properties of on he considers the following lines and areas. Each increasing index number indicates the application of exponentiation, for example , . The lines are without end points, the areas are without boundaries.

These areas are mapped by to (the letters indicate the source area):

Now is simply connected and does not contain 0 (only in the boundary). Hence it is possible to define a holomorphic logarithm on that domain, he defines

on . Which then satisfies

.

And this is the image under , considering that is biholomorphic:

Define and define , then we have and . Define . So one can see that the boundary of consists only of the cyan and violet arcs, hence the points of the real axis in are mapped to the boundary of . This property is used in the final step of Kneser's construction, which follows in my next post.