1. Introduction
Lambert W function is a transcendental function [1] [2] which has applications in many areas of science which include QCD renormalisation, Planck’s spectral distribution law, water movement in soil and population growth [3] - [8] .
Considering the equation
(1)
The Equation (1) can be written as
(2)
Converting the Equation (2) in terms of natural log gives
(3)
Equations ((1)-(3)) have a trivial solution
, but they also have a non- trivial solution.
Figure 1 shows the plot of the function
. The plot indicates that, for any value of the function
in the range of 1 to infinity, it has two different solutions of
. i.e. for any value of
between 1 and infinity, a non-trivial solution of
can be found. The plot also indicates that, at
, there is only one solution
and
(rounded). For any value of y between e and infinity, a solution for x can be found in-between 1 and e.
The solution of Equations ((1)-(3)) can be written in terms of Lambert W function [9] ,
(4)
If
,
and according to Dence [2] ,
, hence
, which is the result obtained graphically and numerically.
Some variations of Equation (1) are:
(5)
(6)
Equation (5) can be written as
(7)
Equation (6) can be written as
(8)
Equation (5) and Equation (6) have trivial solutions of
and
respectively.
2. Non-Trivial Solutions
If
then Equation (1) becomes
and
(rounded) is the nontrivial solution, i.e. 100.13713 = 1.3713 and
If
then Equation (5) and Equation (6) become
and
respectively and their solutions are 13.713 (rounded) and 137.13 (rounded) respectively. These solutions are exactly one and two orders larger than the solution of Equation (1).
Also if
and
, Equation (4) gives
Hence
For the range of integer y values of 4 to 12, the non-trivial solutions for x of Equations ((1), (5) and (6)) were obtained using iterative method. The solutions of x are written in base 10 and in base y (Table 1). Plots of y vs x with x in base 10 and in base y are shown in Figures 2-4 respectively.
3. Conclusions
The non-trivial solutions of Equations ((1), (5) and (6)) written in base y, differ exactly by one order. For y values in the range of 4 to 12, the solutions of Equation (6) written in base y are in the range of 133.33 to 138.99.
When
, the rounded nontrivial solutions for Equation (1), Equation (5) and Equation (6) are 1.3713, 13.713 and 137.13, i.e. 100.13713 = 1.3713,
and
, i.e. for the argument values of 1.3713 and −0.2302, the function values are exactly one order higher. To our knowledge, these results were not reported before.
Table 1. Rounded non-trivial solutions for x of Equations ((1), (5) and (6)) for y values from 4 to 12 are written in base 10 and base y.
Figure 2. Solutions of x in base 10 and in base y for Equation (1) for y values of 1 to 13.
Figure 3. Solutions of x in base 10 and in base y for Equation (5) for y values of 1 to 13.
Figure 4. Solutions of x in base 10 and in base y for Equation (6) for y values of 1 to 13.
The trivial solutions of Equations ((1), (5) and (6)) can be written as 10, 100 and 1000 in base
for any
value.
The non-trivial solution for
of Equation (6), 137.128857 is within 0.1% of the reciprocal value of the atomic fine structure constant
, 137.0359991.
4. Possible Connection to Fine Structure Constant
Allen suggested that
[10] however for the current values of
and
, the relationship is
. Edward Teller suggested ln
, where
is the age of the universe [11] . There could be a connection between Equations ((1) to (8)) and
.