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

and let us pose the following questions. (1) What is the minimum value of f(x)? (2) Can its inverse function be expressed in closed form? (3) Is its indefinite integral known? The function f(x) 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 f(x) = cxt(x). Probably, the most well-known property of f(x) is just its great growth rate. In fact, the rate of increase of f(x) as x → ∞ is greater than the exponential function or the factorial function [3, Chapter I. section 5]


Introduction
Let us consider the following real function of real variable, : R → R: and let us pose the following questions.
(1) What is the minimum value of ( )?
(2) Can its inverse function be expressed in closed form?
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 ( ) = cxt( ). 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 < 0, ( ) 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.

Minimum Value
The usual way to answer the first question is just to solve the equation ( ) = 0; that is, 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, 2 International Journal of Analysis 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 = 0, where the function ( ) does not exist. However, applying L'Hôpital's rule, the following limit is finite: Nevertheless, the right derivative of ( ) at = 0 is infinite: For ̸ = 0, we may rewrite ( ) by using the signum function sgn( ) as Now, for ̸ = 0 and ∈ Z, we have where ( ) is the Heaviside function. Therefore, by applying Euler's formula = cos + sin , we obtain Since ( ) is a real function, the complex part of (11) has to be zero. For > 0, ( ) is never complex, and for < 0 the complex part of ( ) is zero when Therefore, substituting (12) into (11), the function ( ) is given by and its domain is Notice that, despite the fact we have considered the log( ) function as a multivalued function in (10), ( ) is a singlevalued function in (13), because we are considering ( ) as a real function. Figure 1 shows the plot of ( ). According to (13), for < 0, the plot of ( ) lies on the following curves: with a numerable infinite number of points. Notice that + and − signs in (15) occur for even and odd positive integers in (13), respectively.

Minimum
Value of ( ). In order to calculate the minimum value of ( ), for < 0, let us solve the equation Thus, Notice that So, + ( ) has a maximum and − ( ) has a minimum in (17), which agrees with Figure 1. However, −1/ ∉ Dom , so we have to get the best rational approximation − / to −1/ in such a way that − / ∈ Dom . Moreover, since the minimum lies on the − ( ) curve, and must be both odd positive integers. In order to do so, let us consider the sequence where ⌊ ⌋ denotes the floor function. Notice that the numerator and the denominator of (19) are both odd, so ( ) = − / irreducible, with , being odd positive integers. Therefore, ( 0 ) is a sequence of rational numbers for which Defining now the rational number, ∈ Q, Then, the minimum value of ( ) is which is different from (6), as aforementioned.
International Journal of Analysis x Figure 1: Plot of as a real function of real variable.

Inverse Function
About the second question, a closed form expression for the inverse function does not seem to be found in the usual literature (see [2,Chapter 2]). However, by using the Lambert function [5], ( ) is very easily inverted. The Lambert function is defined as the inverse function of and it is implemented in MATHEMATICA by the commands ProductLog or LambertW. The Lambert function is a multivalued function that presents, for real arguments, two branches: 0 ( ) (principal branch) and −1 ( ). Figure 2 shows the plot of both branches.
Let us consider now on a little more general function than (1), but, for simplicity, only for positive arguments; that is, Figure 3 shows the plot of ( ) for different values of . It is easy to prove that which agrees with Figure 3.
Applying now the Lambert function and taking into account (26), we obtain According to Figure 3, notice that, depending on the value of , −1 ( ) sometimes is a double-valued function, so we have two real values of the Lambert function in (29), that is, 0 ( ) and −1 ( ). In this latter case, we have used the notation −1,0 ( ). Also, from Figure 3, we can see that −1 ( ) is a single-valued function for > 1 when < 0 and 4 International Journal of Analysis for ∈ (0, 1) when > 0. Therefore, taking into account (24) and Figure 3, we can consider the following cases.
(i) Consider < 0, (ii) Consider > 0, where we have defined By setting = −1, in (30), we obtain the inverse function of (1) Curiously, (33) is just the closed expression given in [6] for the following power tower: which converges if and only if In order to see this, consider the power tower which converges if and only if [7,8] Taking in (36) the logarithm of both sides and plugging back the ℎ( ) function definition, we obtain log ℎ = ⋅⋅⋅ log = ℎ log .
Performing now the change of variables ℎ = − , we get Now, by using the principal branch of Lambert function, we can solve, for ℎ( ), but, from (34) and (40), we arrive at (33): which, according to (37), converges if and only if that is, the interval given in (35).
In [2,Equation 02.03], −1 ( ) is termed as coupled root function. Since the latter reference is unaware of the closed expression (33), it performs numerically the following limit [2,Equation 02.07]: in order to show that −1 ( ) goes to infinity at a lower rate than logarithms. In fact, (43) is quite easily proved applying (33) and the property so that Performing the change of variables = − log / , we have and expanding the exponential function − , in (47), we obtain Performing now the change of variables 0 ( ) = (i.e., = ), we get where we have set (52) Therefore, applying (52) in (49), we obtain (53) Applying now the property [9, Equation 8.8.1] we may rewrite (53) as In order to compute the lower incomplete gamma function given in (55), we may use [1,Equation 45:4:1] where ( ) is the exponential polynomial and it is given by the power-series expansion of the exponential function by truncation after the th term [1, Equation 26:12:2]: Therefore, we finally obtain where = 0 (− log / ), > 1 and < 0.
Notice that if = 0 and = is a positive integer, then we recover the usual gamma function: We can generalize (61), calculating the integral Expanding in power-series the exponential function given in (64) and integrating term by term, we get Performing now the change of variables +1 = − , we obtain International Journal of Analysis By using now the definition of the upper incomplete gamma function (62), we arrive at In [11,Lemma 10.6], we find a similar expression for the indefinite integral of the power tower Taking = 1 in (67) and using (63), we recover the expression given by [12,Equation 3.342]: Moreover, taking = ±1, in (69), we recover the expressions given for the sophomore's dream [13, pp. 4, 44], discovered in 1697 by Johann Bernoulli, as follows: In order to compute the upper incomplete gamma function given in (67), we may use [1,Equation 45:4:2]: where ( ) is the exponential polynomial (57). Therefore, (67) can be rewritten as Let us now proceed to give another expression for the indefinite integral of by using the results given in (58) and (60). First, let us consider the cases < 0 and > 1, splitting (64) into two summands, as follows: where the first integral in (73) has been substituted by (69) and the second integral can be computed by knowing that ( ) is an increasing function for > 1 when < 0. Indeed, according to Figure 4, we have So, from (73) and (74) and taking into account (58), we get Since the following series is a telescoping series: we can simplify (75), obtaining where = 0 ( log ), > 1 and < 0.
International Journal of Analysis For the case of > 0, according to Figure 5, we have Therefore, substituting (78) in (73) and taking into account (60) and (76), we arrive again at (77), but for > 0. Moreover, the range > 1 can be extended up to the point where is a monotonic increasing or decreasing function. So, according to (24), we can say that > 1/ . Then, collecting all these results, we can conclude that where = 0 ( log ), > 1/ , and > 0.