Let a be fine-structure constant.

a is given by the Hans-de-Vries (HdV) equation:

　a = Γ(a)^2･exp(-ππ/2),

　The Γ(a) factor is a small correction on the Gaussian exponential.

It can be expressed as a so-called radiative series in a.

　　　　　　　　　　　(Hans de Vries, 2004/10/04）


Instead, we suppose Γ can be expressed by finite terms:

　Γ(a) = 1 + 2π･sinh(a/2π) + aa/(2π),

then we get　a = 1／137.035999075193･･･


Ｉ(x) is the modified Bessel function of the first kind. (1/2-th order)

Ｉ(x) = √(2/π)･sinh(x)/√x,

then,　Γ(a) = 1 + (π√a)･Ｉ(a/2π) + aa/(2π),