In this paper, we study the equation of the form of
which can also be written as
. Apart from the trivial solution
x =
y, a non-trivial solution can be expressed in terms of Lambert
W function as
. For
y > e, the solutions of
x are in-between 1 and e. For integer
y values between 4 and 12, the solutions of
x written in base
y are in-between 1.333 and 1.389. The non-trivial solutions of the equations
and
written in base
y are exactly one and two orders higher respectively than the solutions of the equation
. If
y = 10, the rounded nontrivial solutions for the three equations are 1.3713, 13.713 and 137.13,
i.e. 10
0.13713 = 1.3713. Further, ln(1.3713)/1.3713 = 0.2302 and
W(-0.2302) = -2.302. The value 137.13 is very close to the fine structure constant value of 137.04 within 0.1%.
Exponential Equation Lambert W Function Fine Structure Constant1. 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
y x y = x (1)
The Equation (1) can be written as
log y x = x y (2)
Converting the Equation (2) in terms of natural log gives
ln x x = ln y y (3)
Equations ((1)-(3)) have a trivial solution x = y , but they also have a non- trivial solution.
Figure 1 shows the plot of the function ln x x . The plot indicates that, for any value of the function ln x x in the range of 1 to infinity, it has two different solutions of x . i.e. for any value of y between 1 and infinity, a non-trivial solution of x can be found. The plot also indicates that, at y = e , there is only one solution x = e and ln x x = 1 / e = 0.3679 (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] ,
y = W [ − ln ( x ) x ] [ − ln ( x ) x ] (4)
If x = e , y = W [ − 1 e ] [ − 1 e ] and according to Dence [2] , W [ − 1 e ] = − 1 , hence y = e , which is the result obtained graphically and numerically.
Some variations of Equation (1) are:
y x / y 2 = x / y (5)
y x / y 3 = x / y 2 (6)
Equation (5) can be written as
ln x ln y = x y 2 + 1 (7)
Equation (6) can be written as
ln x ln y = x y 3 + 2 (8)
Equation (5) and Equation (6) have trivial solutions of x = y 2 and x = y 3 respectively.
2. Non-Trivial Solutions
If y = 10 then Equation (1) becomes 10 ( x / 10 ) = x and x = 1.3713 (rounded) is the nontrivial solution, i.e. 100.13713 = 1.3713 and
ln x x = ln y y = 0.2302
If y = 10 then Equation (5) and Equation (6) become 10 ( x / 100 ) = x / 10 and 10 ( x / 1000 ) = x / 100 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 x = 1.3713 and y = 10 , Equation (4) gives
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 y = 10 , 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, ln ( 1.3713 ) / 1.3713 = 0.2302 and W ( − 0.2302 ) = − 2.302 , 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.
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
y
Solutions of Equation (1)
Solutions of Equation (5)
Solutions of Equation (6)
In base 10
In base y
In base 10
In base y
In base 10
In base y
12
1.3122
1.389
15.75
13.89
189.0
138.9
11
1.3389
1.380
14.73
13.80
162.0
138.0
10
1.3713
1.371
13.71
13.71
137.1
137.1
9
1.4114
1.363
12.70
13.63
114.3
136.3
8
1.4625
1.355
11.70
13.55
93.6
135.5
7
1.5301
1.350
10.71
13.50
75.0
135.0
6
1.6242
1.343
9.75
13.42
58.5
134.3
5
1.7649
1.340
8.82
13.40
44.1
1340
4
2.0000
1.333..
8.00
13.33..
32.0
133.3
The trivial solutions of Equations ((1), (5) and (6)) can be written as 10, 100 and 1000 in base y for any y value.
The non-trivial solution for x of Equation (6), 137.128857 is within 0.1% of the reciprocal value of the atomic fine structure constant α − 1 , 137.0359991.
4. Possible Connection to Fine Structure Constant
Allen suggested that m e / M p ~ 10 α 2 [10] however for the current values of m e / M p and α , the relationship is m e / M p = 10.227 α 2 . Edward Teller suggested ln T 0 3 / 2 = α − 1 , where T o is the age of the universe [11] . There could be a connection between Equations ((1) to (8)) and α − 1 .
Cite this paper
Gnanarajan, S. (2017) Solutions of the Exponential Equation or and Fine Structure Constant. Journal of Applied Mathematics and Physics, 5, 386-391. https://doi.org/10.4236/jamp.2017.52034
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which can also be written as
. Apart from the trivial solution
x =
y, a non-trivial solution can be expressed in terms of Lambert
W function as
. For
y > e, the solutions of
x are in-between 1 and e. For integer
y values between 4 and 12, the solutions of
x written in base
y are in-between 1.333 and 1.389. The non-trivial solutions of the equations
and
written in base
y are exactly one and two orders higher respectively than the solutions of the equation
. If
y = 10, the rounded nontrivial solutions for the three equations are 1.3713, 13.713 and 137.13,
i.e
. 10
0.13713 = 1.3713. Further, ln(1.3713)/1.3713 = 0.2302 and
W(-0.2302) = -2.302. The value 137.13 is very close to the fine structure constant value of 137.04 within 0.1%.