3010. Calculation of the Exact Value of the Fine Structure Constant, Alpha (α), Using Universal Transcendental Function (UTF)

Published: 10 May 2017 AD
The Feast of St. John of Avila (1562 AD)

Rewritten: 11 July 2012
The Feast of Saint Benedict (543 AD); Saint Pius I (167 AD)

Introduction

Since the fine structure constant is dimensionless, we will use original Equation of the Universal Transcendental Function (UTF) and the two fundamental constants, C0, and C16. This rule may apply also to other dimensionless quantities.

Here is the process of derivation of the exact result of the fine structure constant, alfa, α. The calculation is done in a couple of steps as follows (the value of alfa was calculated using Force FORTRAN, in Double precision, Eclipse FORTRAN and Intel FORTRAN with Quad precision):

The Four-Step Calculation Process

Step One: Calculate the Main Exponent (expm)

The Main Exponent, expm, is calculated using Equation 1:

A = Expm = ( C0 / ( ( C16 / 10. ) (11. / 3. ) ) ) (C16 / 3. )
(Equation 1)

Two transcendental constants are used:

• C0 = 0.986 976 350 384 356 956…
• C16 = 9.999 838 797 804 880 93…

Along with the numbers 3., 10., and 11.

Plugged into Equation 1, this will give exponent m, expm:

A = Expm = 0.957 432 928 678 624 41…

Step Two: Calculate the Transcendental Constant at Point x = (16. + expm)

We have to use the formula for the general transcendental function, which is:

FT ( x ) = ( C0 ) * ( π / e )^x
(Equation 11)

Therefore:

B = FT ( 16. + ( expm = 0.957 432 928 678 624 41… ) )

Substituting the values:

= ( C0 = 0.986 976 350 384 356 956…) * ( π / e = 1.155 727 349 790 92107… )^(16.957 432 928 678 624 41…)

= 11.486 106 001 091 650…

Step Three: Calculate the Second Exponent (exppo)

The third step is to calculate the second exponent, 'partial exponent one', i.e., exppo, as follows using Equation 22:

C = Exppo = 17. / expm = 17. / 0.957 432 928 678 624 41…

= 17.755 812 956 487 822…

Step Four: Calculate the Exact Value of the Fine Structure Constant (α)

Now, we can calculate the exact value of the fine structure constant, α, using Equation 2:

Αlfa( -1. / 2. ) = ( FT ( x ) / 10. )^exppo
(Equation 2)

Where FT (x) equals the result we just calculated, i.e., FT ( 16. + expm )

Putting these into Equation 2, we obtain:

Alfa( - 1. / 2. ) = ( 11.486 106 001 091 650… / 10. )^( 17.755 812 956 487 822…)

= 11.706 237 618 540 260…

To obtain alfa, we have to square the result. This will give the following:

Double precision FORTRAN, alfa:
α( -1.) = 137.035 999 181 727 99…

Results and Comparison

Calculation Results

Double Precision FORTRAN:

• Alfa(-1.) = 137.035 999 181 727 99…
• Reciprocal of alfa, α = 7.29735256407965365…E-003

Quad Precision FORTRAN:

• alfa(-1.) = 137.035 999 181 727 215 672 064 191 303 733…
• Reciprocal of alfa, α = 7.297 352 564 079 694 392 948 518 223 170 759…E-0003

Note: These are calculations in Intel FORTRAN Quad precision from the spring/summer 2017. These agree almost exactly (30 significant digits) with the newest results from the year 2026 posted in article 3010.

Relative Error Analysis

Relative error with experimental result of Nagoya University (2012):

Epsilon, ԑ = 5.638 749 727 x 10⁻¹¹

Comparison with Other Results

Nagoya University Team (2012)

Alfa(-1) = 137.035 999 174 (35)
i.e., (25 ppb)

CODATA (2022)

Alfa(-1) = 137.035 999 177

Simplified Formula

The following components are essential:

Component A:

A = Expm = ( C0 / ( ( C16 / 10. ) ^(11. / 3. ) ) ) ^(C16 / 3. )
(Equation 1)

Component B:

B = FT ( 16. + A ) = ( C0 ) * ( π / e )^(16.+A)
( Equation 11)

Component C:

C = exppo = 17. / A
(Equation 22)

Reciprocal of alfa, α:

Αlfa⁻¹ = [ ( B / 10. )^C ]²
(Equation 33)

Simplified Final Formula

The final result using only the 'A' component and removing 'B' and 'C' components, we obtain the reciprocal of alfa, α:

Alfa⁻¹ = [ (1. / 10. ) * ( C0 ) * ( π / e )^(16. + A) ]^( 34. / A )
(Equation 333)

Where 'A' is equal:

A = [ C0 / ( C16 / 10. )^(11. / 3.) ]^( C16 / 3.)
(Equation 1)

Conclusion

The above formula is excellent for calculating the fine structure constant α. The result obtained in this paper is theoretical, and, it is claimed, exact.

Further Research and References

General Formula Applications

In the old articles with links below, there is a general formula that allows calculation of any number as 'x' in Equation 11 (Transcendental Function). This is important since the coupling constant of the weak force may be obtained.

Additional Resources

The three links to the articles with general formula for obtaining any coupling constant plus more information:

Part I

Derivation of General Formula for Constants of the Cosmology and Quantum Mechanics - Part I

Part II

Derivation of General Formula for Constants of the Cosmology and Quantum Mechanics - Part II

Part III (Graphs)

Graphs and Visual Representations

Historical Dates Referenced

~ 16 May 2017 AD
The Feast of St. Simon Stock (1265 AD); St. Brendan the Navigator (577 AD); St. Gemma Galgani (1903 AD)

Note: The graphs and descriptions in the additional articles are from the past. Please disregard some of the descriptions on them, as they are not proven placements yet.