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

Abstract

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 𝑓𝛽(𝑥) = 𝑥−𝛽𝑥, 𝛽 ∈ R and 𝑥>0, two different expressions in closed form for the inverse function 𝑓−1 𝛽 can be obtained. Also, two different series expansions for the indefinite integral of 𝑓𝛽 and 𝑓−1 𝛽 are derived. The latter does not seem to be found in the literature.