Approximate formulas linking the fine structure constant and the first
The - somehow mysterious - fine structure constant α (see for example Wikipedia),
usually referred by its reciprocal 1/α ~ 137.035999679(94),
is thought to be related with I(0,x), the modified Bessel function of the first kind of order
as you may easily see by a quick overlook on the net; see for example formula (1) at
I have found (using the Inverse Calculator on the web) an approximate formula linking them even
with the - somehow mysterious - (first) Feigenbaum's constant
δ=4.6692016091... (see for example Wolfram MathWorld or Wikipedia)
by this relation
δ ~ (1/α )^1/(I(0,1)sin(1) + Γ (5/12))[=4.669201650...]
which is exact up a relative error less than 1E-8 (or, 1 over 100 millions); and it's inverse, more interesting,
1/α ~ δ^(I(0,1)sin(1) + Γ (5/12))
which is exact up a relative error less than 3E-8.
That seems good enough, for a physical "constant" suspected of slight variations in time.
Bessel functions are notoriously related with goniometric functions and function Γ.
(Pay attention that here above α denotes - as used - the fine structure constant, not
the second Feigenbaum constant 2.502907875... usually denoted by the same
Alessandro Soranzo (2010)