The Fehu Function ᚠ(d) and Exponential Equality
ABSTRACT
This study aims to define a new function ᚠ(d) that satisfies the equation ᚠ(d)(ᚠ(d)+d) = (ᚠ(d)+d)ᚠ(d) and serves as a crossover number for certain exponential properties. The values of the Fehu Function for differences d were calculated using Newton’s method and graphical analysis; materials used were limited to Desmos and the Android calculator app. The results substantiate the proposed properties of ᚠ(d) and show that it properly predicts exponential behavior. The behaviors of the Fehu Function were then analyzed to reach several conjectures, theorems, and identities: the Fehu Equality Identity, Constant Limit Conjecture, Fehu Difference Theorem, and Shortened FEI Corollary.
INTRODUCTION.
This paper explores one of the fundamental functions of mathematics – exponentiation – and requires basic knowledge of this function [1]. To begin, consider the statements 89 > 98 and 21 > 12; in other words, consider the inequality of powers with a difference of 1. For large numbers, n(n+1) > (n+1)n, and for small numbers, (n+1)n > n(n+1). The purpose of the Fehu Function ᚠ(d) is to provide a crossover point where these behaviors change, or rather, to satisfy the equation ᚠ(d)(ᚠ(d)+d) = (ᚠ(d)+d)ᚠ(d). This implies the Fehu Conjugate, or ᚠ̅(d), where ᚠ̅(d) = ᚠ(d)(ᚠ(d)+d) = (ᚠ(d)+d)ᚠ(d) – this expression is termed the Fehu Equality Identity, or FEI (1). The main objective of this paper is to define the Fehu Function and substantiate its properties as a crossover point, as well as to explore identities and theorems regarding the function. It is hypothesized that the Fehu Function allows for the prediction of exponential behavior and properties, provides greater insight into exponentiation as a whole, and introduces a new function for the mathematical world to explore. However, some of these properties are limited to theory and lack formal proofs; additionally, there are limitations in the methods used and the resources available to the investigation. The only theory required is the Fehu Equality Identity: all graphs and theorems are extrapolated from this expression. The Desmos graphing calculator was used for all graphs.
Confirmation of Crossover Property.
To verify the property that ᚠ(d) is a ‘crossover number’, consider the end behaviors of g(x) and f(x), which equal x(x+d) and (x+d)x respectively. At x < 0, both graphs extend into asymptotes and imaginary outputs, which were not analyzed in this paper. At x = 0, f(x) > g(x), and as x→∞, g(x) > f(x). This behavior remains consistent for all positive differences and switches the roles of f(x) and g(x) for negative differences. According to the Intermediate Value Theorem [2], there must be at least one intersection of f(x) and g(x) over the interval [0, ∞); this is hypothesized to be a single case at x = ᚠ(d). Further confirmation was obtained through analysis of the graphs of (x+1)x and x(x+1).
Figure 1 shows a graph of x(x+1) (blue) and (x+1)x (red). Note the single intersection at (2.2931662874…, 15.380632553…), which shows that (x+1)x and x(x+1) are equal for a single input: x = ᚠ(1). Additionally, for x < ᚠ(1), (x+1)x > x(x+1), with the opposite holding for x > ᚠ(1). The end behaviors of the curves, together with the previous analysis, confirm the behavior of the Fehu Function as a crossover point for all positive real x. It is a singular crossover point, as (f-g)(x) is monotonically [3] decreasing when x > ᚠ(1). Note that when d = 0, f(x) = g(x) at all inputs, meaning the Fehu Function fails to apply. The domain of ᚠ(d) is therefore {d|d ≠ 0}.
Extrapolation of a graph of ᚠ(d).
A graph of ᚠ(d) was extrapolated through the FEI (1). By setting ᚠ(d) = y and d = x, the implicit equation (y+x)y=y(y+x) was created. The greater portion of this study is based on observation of this graph (Figure 2).
Calculation of a Fehu value.
A Fehu value for a given difference d can be extrapolated through Newton’s method [4]. By defining p(x) = x(x+d) – (x+d)x, ᚠ(d) can be calculated by finding the zero of p(x). Newton’s method is a recursive algorithm where an initial guess is used to
find the zero of a function. In this case, the initial guess for the calculations was 2. The algorithm is as follows: a = 2-p(2)/p’(2). Repeating this method with a: b = 2-p(a)/p’(a). As the number of terms in the algorithm increases, the output approaches the zero of p(x). Note that as d à 0, this method becomes less accurate. Because of this, all values for 0 < d < 1 have been estimated based on the graph of ᚠ(x). Furthermore, all values for d < 0 have been calculated based on the Fehu Difference Theorem (3). A table of Fehu values is shown below in Table 1.
| Table 1. A table of Fehu and Fehu Conjugate values for given differences. | ||
| d | ᚠ(d) | ᚠ̅(d) |
| -5 | 6.56681357333… | 19.0828140374… |
| -4 | 5.66471428067… | 17.9401548985… |
| -3 | 4.80162766119… | 16.8891710253… |
| -2 | 4 | 16 |
| -1 | 3.29316628741… | 15.3806325543… |
| -0.1 | 2.7669048… | 15.12… |
| -0.01 | 2.723289… | 15.1542705… |
| -0.001 | 2.7187781… | 15.1541910… |
| -0.0001 | 2.718331… | 15.15423711… |
| e (constant included for reference) | 2.7182818… | N/A |
| 0.0001 | 2.718231… | 15.15423711… |
| 0.001 | 2.717781… | 15.1541910… |
| 0.01 | 2.713289… | 15.1542705… |
| 0.1 | 2.669048… | 15.12… |
| 1 | 2.29316628741… | 15.3806325543… |
| 2 | 2 | 16 |
| 3 | 1.80162766119… | 16.8891710253… |
| 4 | 1.66471428067… | 17.9401548985… |
| 5 | 1.56681357333… | 19.0828140374… |
Conjecture: ᚠ(d) \(\rightarrow\) e as d \(\rightarrow\) 0 (Constant Limit Conjecture, 2). This conjecture is derived from the behavior of ᚠ(d) as it approaches zero.
According to Figure 3, the intercept, or limit, falls into the same range as the constant e, supporting the above conjecture. Additionally, Table 1 shows values approaching e as d approaches zero. These figures provide evidence for the Constant Limit Conjecture, as both show that ᚠ(d) à e as d à 0. This limit is different from the classical limit for e [5]. However, a formal proof has yet to be established.
Identity: ᚠ̅(d) = ᚠ(d)(ᚠ(d)+d) = (ᚠ(d)+d)ᚠ(d) (Fehu Equality Identity, 1). This is the very definition of the Fehu function and is the fundamental axiom of all further proofs and extrapolations.
Theorem: ᚠ(-d) – ᚠ(d) = d (Fehu Difference Theorem, 3) and Proof. Begin with the Fehu Equality Identity (1) for a difference of -d. Set a new variable a equal to ᚠ(-d)-d, and substitute this into the expression, giving (a+d)a = a(a+d). This equation has the same form as the Fehu Equality Identity for a difference d (ᚠ(d)(ᚠ(d)+d) = (ᚠ(d)+d)ᚠ(d)); therefore, ᚠ(d) = a = ᚠ(-d)-d. With a slight rearrangement of the equality, the theorem ᚠ(-d)-ᚠ(d) = d is reached. ∎
Corollary: ᚠ(d)ᚠ(-d) = ᚠ(-d)ᚠ(d) (Shortened FEI, 4) and Proof. Begin with the Fehu Inequality Identity (1). Using the Fehu Difference Theorem (4), replace (ᚠ(d)+d) with ᚠ(-d), giving the expression: ᚠ(d)ᚠ(-d) = ᚠ(-d)ᚠ(d). ∎
This paper has answered its inspirational question and confirmed the properties, theorems, and identities of the Fehu Function. Further studies are possible, and indeed, necessary – a graph of Fehu Conjugates has yet to be established, many conjectures lack proofs, and an elegant equation or strategy for calculating ᚠ(d) has yet to be found. Most importantly, the behavior of the Fehu Function given imaginary inputs or explored beyond positive real numbers is unknown. Additionally, there may be a relationship between the Lambert W Function and the Fehu Function, which should be found if possible [6]. Given that this paper lacked significant use of computer calculations, this is the most logical tool to implement in the future. A more advanced graphing system than Desmos may also be effective. In conclusion, the Fehu function is a new, interesting way of describing exponential equality. There are surely many more properties of and uses for this function that have yet to be explored, making the Fehu function ᚠ(d) a mathematical frontier.
ACKNOWLEDGEMENTS.
The Desmos graphing calculator was the main tool used for this graph and should be acknowledged. Additionally, the Android calculator app is effective at finding the Fehu Conjugate values for given Fehu outputs. My family, who encouraged me to research this behavior, should also be acknowledged.
REFERENCES.
- S. A. Mousel, The Exponential Function. MAT Exam Expository Papers 26, 3-15 (2006).
- R. Walter, Principles of Mathematical Analysis (New York: McGraw-Hill, 1976), pp. 93.
- S. Christopher, Monotonic Function. MathWorld – A Wolfram Resource, https://mathworld.wolfram.com/MonotonicFunction.html. Accessed 16 February 2026.
- N. Eagan, G. Hauser, Newton’s Method on a System of Nonlinear Equations. Carnegie Mellon University, 2-3 (2014).
- S. Reichert, e is everywhere. Nat. Phys 15, 982 (2019).
- R.M Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, D.E. Knuth, On the Lambert W Function. Advances in Computational Mathematics 5, 329-359 (1996).
Posted by buchanle on Thursday, May 14, 2026 in May 2026.
Tags: Equality, Exponentiation, Function, Mathematics