A Closed Form for a Spinor Norm Constant

Bohle, Shannon

doi:10.51094/jxiv.5740

##article.authors##

Bohle, Shannon Archivopedia LLC

DOI:

https://doi.org/10.51094/jxiv.5740

キーワード:

complete elliptic integrals、 lemniscate constant、 gamma function、 closed form、 spinor norm

抄録

Takahashi (2024) expresses the norm of a two-component spinor written in fractional-order differential forms as a constant I₀ ≈ 1.774, given both as a double integral over the unit square outside the unit disc and as a one-dimensional integral of complete elliptic integrals, and provides no closed form. The constant is listed as an open problem in the HorizonMath benchmark of unsolved problems with automatic verification. We prove that I₀ = (16π² + Γ(1/4)⁴)/(8√π · Γ(1/4)²), equivalently I₀ = 2E(1/2) − ½K(1/2), and I₀ = (ϖ/2 + π/ϖ)/√2 with ϖ the lemniscate constant. The proof proceeds from the double integral and is elementary: a trigonometric substitution reduces the domain to a triangle, a rotation of coordinates separates the integrand over a square, and the four resulting one-dimensional integrals consist of two elementary evaluations and two beta values. No elliptic-integral theory is used.

利益相反に関する開示

The author declares no conflicts of interest. This research received no external funding.

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