The fine-structure constant α  is a constant in physics that plays a fundamental role in the electromagnetic interaction. It is a dimensionless constant, defined as: (1)
being q the elementary charge, ε0 the vacuum permittivity, h the Planck constant and c the speed of light in vacuum. The value shown in (1) is according CODATA 2014 .
In this paper, it will be explained that the fine-structure constant is one of the roots of the following equation: (2)
being e the mathematical constant e (the base of the natural logarithm). One of the solutions of this equation is: (3)
This means that it is equal to the CODATA value in nine decimal digits (or the seven most significant ones if you prefer). And therefore, the difference between both values is: (4)
This coincidence is higher in orders of magnitude than the commonly accepted necessary to validate a theory towards experimentation.
As the cosine function is periodical, the Equation (2) has infinite roots and could seem the coincidence is just by chance. But as it will be shown in the paper, the separation among the different solutions is sufficiently high to disregard this possibility.
It will also be shown that another elegant way to show Equation (2) is the following (being i the imaginary unit): (5)
having of course the same root (3). The possible meaning of this other representation (5) will be explained.
Bouchendira, R., Cladé, P., Guellati-Khélifa, S., Nez, F. and Biraben, F. (2010) New Determination of the Fine-Structure Constant and Test of the Quantum Electrodynamics. Physical Review Letters, 106, Article ID: 080801.
Euler, L. (1748) Chapter 8: On Transcending Quantities Arising from the Circle of Introduction to the Analysis of the Infinite, 214, Section 138. Bruce, I., Trans.