Winding Number Weights at Elliptic Points

Explicit computation of generalized winding numbers of the fundamental domain boundary around the elliptic points i, ρ, ρ+1.

Main Results

• gWN_fdBoundary_H_at_i — gWN = -1/2 at i
• gWN_fdBoundary_H_at_rho — gWN = -1/6 at ρ
• gWN_fdBoundary_H_at_rho_plus_one — gWN = -1/6 at ρ+1
• effectiveWinding_rho_eq_neg_gWN — 1/3 = -(gWN(ρ) + gWN(ρ+1))
• effectiveWinding_i_eq_neg_gWN — 1/2 = -gWN(i)

theorem effectiveWinding_rho_eq_neg_gWN (H : ℝ) (hH : √3 / 2 < H) : ↑1 / 3 = -(generalizedWindingNumber' (fdBoundaryH H) 0 5 ellipticPointRho + generalizedWindingNumber' (fdBoundaryH H) 0 5 ellipticPointRhoPlusOne)

theorem effectiveWinding_i_eq_neg_gWN (H : ℝ) (hH : 1 < H) : ↑1 / 2 = -generalizedWindingNumber' (fdBoundaryH H) 0 5 Complex.I