1. Introduction
Lambert W function has applications in science [1] [2] [3] [4], especially in physics [5] [6]. The Lambert W function has applications in quantum statistics, and it is used to derive Wien’s displacement law in connection with the Planck’s black body spectral distribution [7] [8] [9], but it has not been used to describe the spectral distribution. Here we present an application to determine the frequencies in the Planck’s black body spectral distribution, for a specific intensity much less than maximum intensity, at a temperature.
Euler found the solution for the equation
in the form of the Lambert W function in the 18th century [1]. Recently, an exponential form of this equation was used with iterative technique to find solutions [10] [11], but the iteration progression towards convergence has not been investigated. Here we investigate the iteration progression and found solutions for the Euler’s equation for a large range of numbers.
The Lambert W function is defined by
. For real numbers, when < 0,
is a double valued function.
In the region
, it is denoted as
, and in the region
, it is denoted as
.
The plot
vs
is shown in Figure 1, and the plot
vs
for
, is shown in Figure 2. Figure 2 displays more detail description of the Lambert W function in the region 0 to −∞.
2. Lambert W Function and Planck’s Radiation Law
In the Planck’s radiation law, the spectral radiance in terms of frequency [7] is given by
(2.1)
The frequency
corresponds to the maximum intensity [8] [9] is given by
(2.2)
In the Planck’s radiation curve at a temperature, for any one intensity below the maximum intensity, two different frequencies can be found. Consider
and
are two frequencies correspond to one intensity.
Figure 2.
vs
plot.
This implies
(2.3)
(2.4)
If
,
.
The Equation (2.4) can be written as
(2.5)
For
, region where Raleigh-Jean’s law applies, the Equation (2.5) can be written as
(2.6)
Let the ratio
.
The Equation (2.6) can be written as
(2.7)
(2.8)
(2.9)
and
(2.10)
This new Equation (2.10) provides the solutions for the frequencies at which the intensities are equal, with the conditions
and
. This equation is in the same form as the Equation (2.2) for the
.
Table 1 gives the calculated values for the intensity ratio for the frequencies
and
. The ratio is close to one for
, as expected.
3. Euler’s Transcendental Equation and Lambert W Function
The solution for the equation
is given by
derived by Euler in the 18th century [1] [10] [11].
Theorem: The solutions for the series of exponential equations
is given by
.
Proof:
One form of analytical solutions for the series of exponential equations was derived previously [10] [11].
(3.1)
The solutions derived previously:
, trivial solutions (3.2)
and
Table 1. Intensity ratio for different r values with other functions.
, non-trivial solutions (3.3)
The non-trivial solutions can be refined further.
The Equation (3.1) can also be written as
(3.4)
Rearranging the Equation (3.4)
(3.5)
Using Equation (3.5), the solution in Equation (3.3) can be written as:
(3.6)
Rearranging the Equation (3.6)
(3.7)
Hence the solution for the Equation (3.1) can be written as
(3.8)
If
, the Equation (3.1) becomes
(3.9)
i.e.
(Euler’s equation).
The solution is
(3.10)
The
is maximum at
. For
, the non-trivial solutions are in terms of
and for
, the non-trivial solutions are in terms of
.
4. Numerical Calculation
The numerical values of the function in Equation (3.10) were calculated using the Equation (3.9), utilizing the iterative technique. The iteration progresses are shown in Figure 3 for few
values. For
, the iteration converges to the non-trivial solution. For
, the iteration converges to the trivial solution. At
,
, the trivial and the nontrivial solutions are equal.
The non-trivial solutions in the range of
were determined, using the
symmetry in Equation (3.9). For
, even when the seed value close to the non-trivial solution the iteration is unstable (Figure 4).
The numerical values of the function
are given in Table 2.
Figure 3. Iteration steps for non-trivial (nt) solutions for
values from 4 to 15.
Figure 4. Iteration steps for trivial (t) and non-trivial (nt) solutions for
values of 1.5, 2 and 3.
Table 2. Calculated values of function
, given in terms of
and
, depending on the range.
Using the values of the function in Table 2, the plots of the function
, for n = −2, −1, 0, 1 and 2 are shown in Figure 5. Plot of
calculated using the vales in Table 2 and the comparison plot of
are shown in Figure 6.
Figure 5. Plots of function
for n of −2, −1, 0, 1, and 2.
For different values of n,
and
in Equation (3.8), using Table 2, following numerical equations can be obtained
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
Figure 6. Plots of
and
.
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
5. Numerical Coincidences
Consider the Equations (4.9) and (4.16),
These solutions are unique. Numerical coincidences for these numbers with physical constants are given below:
The dimensionless electromagnetic fine structure constant
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
The dimensionless gravitational fine structure constant defined using electron mass can be written as
(5.6)
For convenience hereafter
will be referred as
(5.7)
(5.8)
(5.9)
The numerical values of
and
are close and it was suggested that they are related [12].
In Equation (4.9), if 10 is replaced with
, the equation becomes
(5.10)
6. Conclusion
In the Planck’s radiation law equation, for a specific temperature and intensity, the frequencies will be given by
and
, with conditions
, and
. The numerical calculations of the intensity at these frequencies validated the equations.
A new form of solution for the Euler’s equation
was derived in the form of the Lambert W function as,
, and the corresponding solutions for the series of exponential equations. Interesting numerical equations were derived and coincidences with electromagnetic fine structure constant were indicated.
“God used beautiful mathematics in creating the world” quote by Paul Dirac.