Some Remarks on the Self-Exponential Function: Minimum Value, Inverse Function, and Indefinite Integral

Considering the function xx as a real function of real variable, what is its minimum value? Surprisingly, the minimum value is reached for a negative value of x. Furthermore, considering the function fβx=x-βx, β∈R and x>0, two different expressions in closed form for the inverse function fβ-1 can be obtained. Also, two different series expansions for the indefinite integral of fβ and fβ-1 are derived. The latter does not seem to be found in the literature