03/04/2008, 11:17 AM

As I just read in Knoebel's Exponential Reiterated there is even a parametrization of the curve , already given by Goldbach:

Knoebel considers the equation which is equivalent to which means that the fixed points and have the same base. So . The parametrization is:

and .

We can easily verify that this indeed satisfy :

For example for we get our famous fixed points and

There is lots of other interesting stuff in Knoebel's article, but read it yourself

For our consideration here let:

and so that . So I wonder whether we can express ( and are indeed bijective) with the Lambert W function. Any ideas?

Knoebel considers the equation which is equivalent to which means that the fixed points and have the same base. So . The parametrization is:

and .

We can easily verify that this indeed satisfy :

For example for we get our famous fixed points and

There is lots of other interesting stuff in Knoebel's article, but read it yourself

For our consideration here let:

and so that . So I wonder whether we can express ( and are indeed bijective) with the Lambert W function. Any ideas?