Solutions for Series of Exponential Equations in Terms of Lambert-W Function and Fundamental Constants

Series of exponential equations in the form of were solved graphically, numerically and analytically. The analytical solution was derived in terms of Lambert-W function. A general numerical solution for any y is found in terms of n or in base y. A solution is close to the fine structure constant. The equation which provided the solution as the fine structure constant was derived in terms of the fundamental constants.

1. Introduction

Exponential equations are widely used in natural and social sciences. In this paper, we considered series of exponential equations and solved them graphically, numerically, and analytically in terms of Lambert-W function. One equation connected to the fine structure constant, was derived in terms of the fundamental constants and led to a new equation. The Lambert-W function for real variables is defined by the equation $W\left(x\right)\mathrm{exp}\left[W\left(x\right)\right]=x$     and it has applications in Planks spectral distribution law   , QCD renormalization  , solar cells  , bio-chemical kinetics  , optics  , population growth and water movement in soil  .

Considering the series of exponential equations defined by the following equation

$x={y}^{n}{y}^{\left(\frac{x}{{y}^{n+1}}\right)}$ (1.1)

where x, y, n are real variables.

Taking logy on both sides of the Equation (1.1)

${\mathrm{log}}_{y}x=n+\frac{x}{{y}^{n+1}}$ (1.2)

Converting the Equation (1.2) to natural logarithm

$\frac{\mathrm{ln}x}{\mathrm{ln}y}=\frac{x}{{y}^{n+1}}+n$ (1.3)

The trivial solution of the Equations (1.1) to (1.3) is

$x={y}^{n+1}$ (1.4)

In this paper, we are focusing on the non-trivial solutions.

For n = 2, 1, 0, −1, −2, the Equations (1.1) and (1.3) become:

$x={y}^{2}{y}^{\frac{x}{{y}^{3}}}\text{or}\frac{\mathrm{ln}x}{\mathrm{ln}y}=\frac{x}{{y}^{3}}+2$ (1.5)

$x=y{y}^{\frac{x}{{y}^{2}}}\text{or}\frac{\mathrm{ln}x}{\mathrm{ln}y}=\frac{x}{{y}^{2}}+1$ (1.6)

$x={y}^{\frac{x}{y}}\text{or}\frac{\mathrm{ln}x}{\mathrm{ln}y}=\frac{x}{y}\text{or}{x}^{y}={y}^{x}$ (1.7)

$x={y}^{-1}{y}^{x}\text{or}\frac{\mathrm{ln}x}{\mathrm{ln}y}=x-1$ (1.8)

$x={y}^{-2}{y}^{xy}\text{or}\frac{\mathrm{ln}x}{\mathrm{ln}y}=xy-2$ (1.9)

2. Graphical Solutions

If y =10, the Equations (1.5) to (1.9) become

$x={10}^{2}×{10}^{\frac{x}{{10}^{3}}}$ (2.1)

$x=10×{10}^{\frac{x}{{10}^{2}}}$ (2.2)

$x={10}^{\frac{x}{10}}$ (2.3)

$x={10}^{-1}×{10}^{x}$ (2.4)

$x={10}^{-2}×{10}^{10x}$ (2.5)

The curves and the straight line to obtain the graphical solutions of the Equations (2.1) to (2.5) are shown in Figure 1. The intercepts of the curves and the straight line indicate the solutions.

Figure 1. Plots of the functions to obtain the graphical solutions for the Equations (2.1) to (2.5).

The intersecting points of 0.1, 1, 10, 100 and 1000 are the trivial solutions and the intersecting points at around 0.0137, 0.137, 1.37, 13.7 and 137 are the non-trivial solutions.

The non-trivial solutions imply the following equations:

${10}^{0.1371}=1.371$ (2.6)

$\frac{\mathrm{ln}1.371}{1.371}=\frac{\mathrm{ln}10}{10}=0.2302$ (2.7)

${10}^{1.371}={1.371}^{10}=23.5$ (2.8)

3. Numerical Solutions

Higher precision non-trivial numerical solutions were obtained for the series of equations $x={y}^{n}{y}^{\left(\frac{x}{{y}^{n+1}}\right)}$ using the iterative technique for n = 2, 1.5, 1, 0.5, 0, −0.5, −1, −2 and 1 ≤ y ≤ 15 (Table 1). The iterations do not converge on non-trivial solutions for y < e, and solutions in this range were obtained by trial and error.

The solutions in Table 1 for n = −2, −1, 0, 0.5, 1, 2 are plotted as x vs y with x axis in log scale (Figure 2). Sharp turning points in the plots are observed for y values in the range of 1 to 2.

4. Analytical Solution

Consider the Equation (1.3)

$\frac{\mathrm{ln}x}{\mathrm{ln}y}=\frac{x}{{y}^{n+1}}+n$

Table 1. Non-trivial numerical solutions for the series of equations $x={y}^{n}{y}^{\left(\frac{x}{{y}^{n+1}}\right)}$ .

Figure 2. Plots of x vs y for the series of equations, $x={y}^{n}{y}^{\left(\frac{x}{{y}^{n+1}}\right)}$ for n = 2, 1, 0, 0.5, −1 and −2.

Let

$t=-\mathrm{ln}x$

Then (1.3) becomes

$\frac{-t}{\mathrm{ln}y}=\frac{{\text{e}}^{-t}}{{y}^{n+1}}+n$

$\left(t+n\mathrm{ln}y\right){\text{e}}^{t}=\frac{-\mathrm{ln}y}{{y}^{n+1}}$

$\left(t+n\mathrm{ln}y\right){\text{e}}^{t+n\mathrm{ln}y}=\frac{-{\text{e}}^{n\mathrm{ln}y}\mathrm{ln}y}{{y}^{n+1}}$

$\left(t+n\mathrm{ln}y\right){\text{e}}^{t+n\mathrm{ln}y}=\frac{-\mathrm{ln}y}{y}$

$\left(t+n\mathrm{ln}y\right)=W\left(\frac{-\mathrm{ln}y}{y}\right)$

Substituting −lnx for t

$\left(-\mathrm{ln}x+n\mathrm{ln}y\right)=W\left(\frac{-\mathrm{ln}y}{y}\right)$

Using the Equation (1.3)

$-\frac{x\mathrm{ln}y}{{y}^{n+1}}=W\left(\frac{-\mathrm{ln}y}{y}\right)$

Hence the solution to Equation (1.3) is

$x=\frac{W\left(\frac{-\mathrm{ln}y}{y}\right)}{\left(-\frac{\mathrm{ln}y}{{y}^{n+1}}\right)}=\frac{{y}^{n}W\left(\frac{-\mathrm{ln}y}{y}\right)}{\left(-\frac{\mathrm{ln}y}{y}\right)}$ (3.1)

If n = 0, the Equation (1.1) $x={y}^{n}{y}^{\left(\frac{x}{{y}^{n+1}}\right)}$ becomes Equation (1.7) $x={y}^{\left(\frac{x}{y}\right)}$ .

Using the solution in the Equation (3.1), the analytical solution in terms of Lambert-W function is

$x=\frac{W\left(-\frac{\mathrm{ln}\left(y\right)}{y}\right)}{\left(-\frac{\mathrm{ln}\left(y\right)}{y}\right)}$ (3.2)  

In Equation (3.2), if $y=\text{e}$ , $x=\frac{W\left(-\frac{1}{\text{e}}\right)}{\left(-\frac{1}{\text{e}}\right)}.$

But $W\left(\frac{1}{\text{e}}\right)=-1$  .

Hence $x=\text{e}$ , the result in Table 1.

If n =0 and y =2 in Equation (3.2), the solutions in Table 1 and Equation (1.7) gives

${2}^{4}={4}^{2}$ and $\frac{\mathrm{ln}2}{2}=\frac{\mathrm{ln}4}{4}=-0.346.$ (3.3)

Equation (3.1) gives

$4=\frac{W\left(-\frac{\mathrm{ln}\left(2\right)}{2}\right)}{\left(-\frac{\mathrm{ln}\left(2\right)}{2}\right)}=\frac{W\left(-0.346\right)}{-0.346}$

$2=\frac{W\left(-\frac{\mathrm{ln}\left(4\right)}{4}\right)}{\left(-\frac{\mathrm{ln}\left(4\right)}{4}\right)}=\frac{W\left(-0.346\right)}{-0.346}$

$W\left(-0.346\right)$ is double valued with−0.693 and −1.386.

If we substitute the solutions for n = 0 and y = 10 from Table 1 to Equation (3.2);

$W\left(-\frac{\mathrm{ln}\left(1.37129\right)}{1.37129}\right)=10\left(-\frac{\mathrm{ln}\left(1.37129\right)}{1.37129}\right)$ (3.4)

$W\left(-0.2302\right)=-2.302$ (3.5)

Since x and y are symmetric in Equation (1.7)

$W\left(-\frac{\mathrm{ln}\left(10\right)}{10}\right)=1.371289\left(-\frac{\mathrm{ln}\left(10\right)}{10}\right)$ (3.6)

$W\left(-0.2302\right)=-0.3157$ (3.7)

The W(x) has two real values for $-1/\text{e}\le x<0$  .

If n = −1, the Equation (1.1) $x={y}^{n}{y}^{\left(\frac{x}{{y}^{n+1}}\right)}$ becomes Equation (1.8) $x={y}^{-1}{y}^{x}$ or $xy={y}^{x}$ .

Using the solution in the Equation (3.1), the analytical solution in terms of the Lambert-W function is

$x=\frac{W\left(-\frac{\mathrm{ln}y}{y}\right)}{-\mathrm{ln}y}$ (3.8) 

If $y=\text{e},x=\frac{W\left(-\frac{1}{\text{e}}\right)}{-\mathrm{ln}\text{e}}.$

But $W\left(\frac{-1}{e}\right)=-1$ , Hence x = 1, the result in Table 1.

In Table 1, for any value of n $y=\text{e},x=\text{e}×{\text{e}}^{n}$ , the trivial and nontrivial solutions coincide.

$y=10,x=1.371289×{10}^{n}$

Using the solution in Equation (3.1), for any y the solution x can be written as

$x\left(n=0\right)×{y}^{n}$ (3.9)

Plots of lnx vs n shown in Figure 3 are linear as expected from Equation (3.1).

The lnx vs n lines for different y values are crossing near the point (0.5, 1.4). This indicates the solutions for n = 0.5 have little dependency on y for y ≥ e. This is also evident in the numerical results for n = 0.5 in Table 1 and in the plot of $x={y}^{0.5}{y}^{\frac{x}{1.5}}$ in Figure 2.

5. Solutions x in Base y

The solutions x in Table 1 can be written in base y, (xy) to indicate the general pattern.

For any valued of n, xy can be written as

${x}_{y}\left(n=0\right)×{10}^{n}$ (5.1)

For n = 2, the solutions written in base y, xy shown in Table 2.

For y > 11, the xy are written using the hex notation.

There is a sharp change in the value of the xy at y = 4.

For n = 2, plot xy vs y, for 5 ≤ y ≤ 11 is shown in Figure 4.

6. Connection to the Fine Structure Constant

In Equation (1.1), when n = 2 and y = 10, the equation becomes $x={10}^{2}×{10}^{\frac{x}{{10}^{3}}}$ and the solution is

Figure 3. Plots of lnx vs n.

Table 2. Solutions x in base y(xy).

Figure 4. Plot of xy vs y for n = 2.

$x=\frac{{10}^{2}W\left(\frac{-\mathrm{ln}10}{10}\right)}{\left(-\frac{\mathrm{ln}10}{10}\right)}=137.129$ (6.1)

The solution 137.129 is close to the inverse of the fine structure constant 137.036  -  which is dimensionless.

The inverse of the fine structure constant ${\alpha }^{-1}$ is given by the expression

${\alpha }^{-1}=\frac{4\text{π}{\epsilon }_{o}\hslash c}{{e}^{2}}=137.036$ (6.2)

where;

$\hslash =1.0545718×{10}^{-34}\text{\hspace{0.17em}}\text{J}\cdot \text{s}$ ,reduced Planck constant;

$c=2.99792458×{10}^{8}\text{\hspace{0.17em}}\text{m}\cdot {\text{s}}^{-1}$ , speed of light in vacuum;

${\epsilon }_{0}=8.854187817×{10}^{-12}\text{\hspace{0.17em}}\text{F}\cdot {\text{m}}^{-1}$ , electric constant;

$e=1.6021766208×{10}^{-19}\text{\hspace{0.17em}}\text{C}$ , elementary charge;

${\alpha }^{-1}$ , dimensionless constant  .

In a recent publication Eaves  suggested an equation relating G and α;

$\frac{\alpha {q}^{2}}{8{\text{π}}^{2}G{m}_{e}^{2}}\approx \mathrm{exp}\left(\frac{2}{3\alpha }\right)$ (6.3)

where;

$G=6.67408×{10}^{-11}\text{\hspace{0.17em}}{\text{m}}^{3}\cdot {\text{kg}}^{-1}\cdot {\text{s}}^{-2}$ , gravitational constant;

${m}_{e}=9.10938356×{10}^{-31}\text{\hspace{0.17em}}\text{kg}$ , electron mass.

${q}^{2}=\frac{{e}^{2}}{4\pi {\epsilon }_{0}}$

By substituting the expression for α in Equation (6.3) we get

$\frac{{e}^{4}}{32{\text{π}}^{3}{\epsilon }_{0}^{2}G{m}_{e}^{2}\hslash c}\approx \mathrm{exp}\left(\frac{2}{3\alpha }\right)$ (6.4)

Using Equation (6.2), the Equation (6.4) becomes

${\alpha }^{-1}\approx \frac{64{\text{π}}^{4}{\epsilon }_{0}^{3}{\hslash }^{2}{c}^{2}G{m}_{e}^{2}}{{e}^{6}}\mathrm{exp}\left(\frac{{\alpha }^{-1}}{1.5}\right)$ (6.5)

Substituting numerical values for the pre-exponent,

${\alpha }^{-1}\approx 1.59947×{10}^{-27}\mathrm{exp}\left(\frac{{\alpha }^{-1}}{1.5}\right)$ (6.6)

${\alpha }^{-1}\approx 1.59947×{10}^{-27}×{10}^{\left(\frac{{\alpha }^{-1}}{3.4538}\right)}$ (6.7)

By taking the power of (1/289.5) on both sides of the Equation (6.8) and writing the equation for ${\alpha }^{-1}$ yields

${\alpha }^{-1}\approx 106.6×{10}^{\left(\frac{{\alpha }^{-1}}{1000}\right)}$ (6.8)

The Equation (6.8) is approximately the same as the equation $x={10}^{2}×{10}^{\frac{x}{{10}^{3}}}$ . The only difference is the 102 is 106.6 in Equation (6.8). But the Equation (6.8) based on the Equation (6.3) is only an approximate equation.

The value $1.59947×{10}^{-27}$ in Equation (6.6) is approximately equal to the ${\alpha }_{G}^{1/1.5},$ ${\alpha }_{G}$ defined by Jentschura  .

${\alpha }_{G}=\frac{G{m}_{e}{m}_{p}}{\hslash c}=3.21×{10}^{-42}$ (6.9)

Hence the Equation (6.7) can be written as

${\alpha }^{-1}\approx {\alpha }_{G}^{1/1.5}\mathrm{exp}\left(\frac{{\alpha }^{-1}}{1.5}\right)$ (6.10)

7. Conclusions

An equation in the form of $x={y}^{n}{y}^{\left(\frac{x}{{y}^{n+1}}\right)}$ was solved graphically, numerically and analytically.

The plots of numerical solution x vs y indicate sharp turning points for y values in-between 1 to 2.

The analytical solution was found in terms of Lambert-W function as

$x=\frac{{y}^{n}W\left(\frac{-\mathrm{ln}y}{y}\right)}{\left(-\frac{\mathrm{ln}y}{y}\right)}$

The numerical solutions can be written as $x\left(n=0\right)×{y}^{n}.$

The numerical solutions can also be written in base y as ${x}_{y}\left(n=0\right)×{10}^{n}$ . For $y\ge 5{x}_{y}\left(n=0\right)$ is a universal number approximately equal to 1.37.

If n = 2 and y = 10, the solution $x=\frac{{10}^{2}W\left(\frac{-\mathrm{ln}10}{10}\right)}{\left(-\frac{\mathrm{ln}10}{10}\right)}=137.129$ (rounded) is close to the inverse of the fine structure constant value, 137.036.

The equation $x={10}^{2}×{10}^{\frac{x}{{10}^{3}}}$ which gives the solution close to the fine structure constant can be derived from the equation $\frac{\alpha {q}^{2}}{8{\text{π}}^{2}G{m}_{e}^{2}}\approx \mathrm{exp}\left(\frac{2}{3\alpha }\right)$ suggested by Eaves.

The derivation resulted in an equation ${\alpha }^{-1}\approx {\alpha }_{G}^{1/1.5}\mathrm{exp}\left(\frac{{\alpha }^{-1}}{1.5}\right)$ .

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Gnanarajan, S. (2018) Solutions for Series of Exponential Equations in Terms of Lambert-W Function and Fundamental Constants. Journal of Applied Mathematics and Physics, 6, 725-736. doi: 10.4236/jamp.2018.64065.

  Corless, R.M., et al. (1996) On the LambertW Function. Advances in Computational Mathematics, 5, 329-359. https://doi.org/10.1007/BF02124750  Dence, T.P. (2013) A Brief Look into the Lambert W Function. Applied Mathematics, 4, 887. https://doi.org/10.4236/am.2013.46122  Kalman, D. (2001) A Generalized Logarithm for Exponential-Linear Equations. The College Mathematics Journal, 32, 2-14.https://doi.org/10.1080/07468342.2001.11921844  Fukushima, T. (2013) Precise and Fast Computation of Lambert W-Functions without Transcendental Function Evaluations. Journal of Computational and Applied Mathematics, 244, 77-89. https://doi.org/10.1016/j.cam.2012.11.021  Valluri, S.R., et al. (2009) The Lambert W Function and Quantum Statistics. Journal of Mathematical Physics, 50, Article Id: 102103. https://doi.org/10.1063/1.3230482  Valluri, S.R., Jeffrey, D.J. and Corless, R.M. (2000) Some Applications of the Lambert W Function to Physics. Canadian Journal of Physics, 78, 823-831.  Scott, T.C., Mann, R. and Martinez Ii, R.E. (2006) General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function A Generalization of the Lambert W Function. Applicable Algebra in Engineering, Communication and Computing, 17, 41-47. https://doi.org/10.1007/s00200-006-0196-1  Jain, A. and Kapoor, A. (2004) Exact Analytical Solutions of the Parameters of Real Solar Cells Using Lambert W-Function. Solar Energy Materials and Solar Cells, 81, 269-277. https://doi.org/10.1016/j.solmat.2003.11.018  Golicnik, M. (2012) On the Lambert W Function and Its Utility in Biochemical Kinetics. Biochemical Engineering Journal, 63, 116-123. https://doi.org/10.1016/j.bej.2012.01.010  Kitis, G. and Vlachos, N. (2013) General Semi-Analytical Expressions for TL, OSL and Other Luminescence Stimulation Modes Derived from the OTOR Model Using the Lambert W-Function. Radiation Measurements, 48, 47-54. https://doi.org/10.1016/j.radmeas.2012.09.006  Barry, D., et al. (1993) A Class of Exact Solutions for Richards’ Equation. Journal of Hydrology, 142, 29-46. https://doi.org/10.1016/0022-1694(93)90003-R  Weisstein, E.W. (2002) Lambert W-Function.  Gnanarajan, S. (2017) Solutions of the Exponential Equation yx/y = x or lnx/x = lny/y and Fine Structure Constant. Journal of Applied Mathematics and Physics, 5, 386. https://doi.org/10.4236/jamp.2017.52034  Kinoshita, T. (1996) The Fine Structure Constant. Reports on Progress in Physics, 59, 1459. https://doi.org/10.1088/0034-4885/59/11/003  Dirac, P.A. (1937) The Cosmological Constants. Nature, 139, 323. https://doi.org/10.1038/139323a0  Bouchendira, R., et al. (2011) New Determination of the Fine Structure Constant and Test of the Quantum Electrodynamics. Physical Review Letters, 106, Article ID: 080801. https://doi.org/10.1103/PhysRevLett.106.080801  Hammer, E. (2006) Physical and Mathematical Meaning of the Alpha Constant, Einstein’s Equation, and Planck Dimensions. Industry Applications Conference, Tampa, 8-12 October 2006. https://doi.org/10.1109/IAS.2006.256848  Kragh, H. (2003) Magic Number: A Partial History of the Fine-Structure Constant. Archive for History of Exact Sciences, 57, 395-431.  Sandvik, H.B., Barrow, J.D. and Magueijo, J. (2002) A Simple Cosmology with a Varying Fine Structure Constant. Physical Review Letters, 88, Article ID: 031302. https://doi.org/10.1103/PhysRevLett.88.031302  Jentschura, U.D. (2014) Fine-Structure Constant for Gravitational and Scalar Interactions. Physical Review A, 90, Article ID: 022112. https://doi.org/10.1103/PhysRevA.90.022112  Jentschura, U.D. and Nándori, I. (2014) Attempts at a Determination of the Fine-Structure Constant from First Principles: A Brief Historical Overview. The European Physical Journal H, 39, 591-613. https://doi.org/10.1140/epjh/e2014-50044-7  Mohr, P.J., Newell, D.B. and Taylor, B.N. (2016) CODATA Recommended Values of the Fundamental Physical Constants: 2014. Journal of Physical and Chemical Reference Data, 45, Article ID: 043102. https://doi.org/10.1063/1.4954402  Eaves, L. (2018) A Model to Inter-Relate the Values of the Quantum Electrodynamic, Gravitational and Cosmological Constants. 