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

Considering the function as a real function of real variable, what is its minimum value? Surprisingly, the minimum value is reached for a negative value of . Furthermore, considering the function , and , two different expressions in closed form for the inverse function can be obtained. Also, two different series expansions for the indefinite integral of and are derived. The latter does not seem to be found in the literature. 1. Introduction Let us consider the following real function of real variable, : and let us pose the following questions.(1)What is the minimum value of ?(2)Can its inverse function be expressed in closed form?(3)Is its indefinite integral known? The function is termed as self-exponential function in [1, Section 26:14] and coupled exponential function in [2, Equation 01.20], using in the latter the notation . Probably, the most well-known property of is just its great growth rate. In fact, the rate of increase of as is greater than the exponential function or the factorial function [3, Chapter I. section 5] Regarding the domain of , in [1, Section 26:14], is only defined as a real function for positive values of , and [2, p. 10] states that, for arguments less than zero, is complex except for negative integers. However, [4] says that, for , is only defined if can be written as , where and are positive integers and is odd. We will use this fact later on in order to answer the first question. This paper is organized such that each section is devoted to each of the questions raised above. 2. Minimum Value The usual way to answer the first question is just to solve the equation ; that is, so Since then (4) is a local minimum. Moreover, since there are no more local extrema and is a smooth function, (4) is the global minimum; thus, Nonetheless, this reasoning fails, because it does not take into account negative values of . Therefore, we need to define first the domain of for negative values of . Despite the fact that this is essentially done in [4], in order to answer the first question, we provide the following derivation. 2.1. Domain of Let us consider first the case , where the function does not exist. However, applying L’H？pital’s rule, the following limit is finite: Nevertheless, the right derivative of at is infinite: For , we may rewrite by using the signum function as Now, for and , we have where is the Heaviside function. Therefore, by applying Euler’s formula , we obtain Since is a real function, the complex part of (11) has to be zero. For , is never complex, and for the complex part of is zero when Therefore, substituting