Daniel Wrote:I am interested to hear what people consider needs to be proven to establish tetration for real and complex numbers.

Well, for me the following assertions need to be proved.

- Do the coefficients in Andrew's solution converge?

- Do the series made up of this coefficients converge? What is the convergence radius?

- Does the natural development at every point define the same analytic function? Is the piecewise defined function analytic?

- Is Andrew's solution for bases identical with the solution obtained by regular development at the lower fixed point?

- Is the solution analytic with respect to the base as argument (especially at the base )?

I call it Andrew's solution merely to give it a common name, though the idea was published before him by Peter Walker.