2018/06/20 by F. D. M. Haldane · 32 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #Elliptic function #Invariant (physics) #Lattice (music) #Quasiperiodicity #Sigma #Theta function #Torus #Wave function #Weierstrass functions #cond-mat.str-el #math-ph #math.MP
paper · pdf · doi:10.1063/1.5042618
published in Journal of Mathematical Physics 59(7) (American Institute of Physics) · 5 pages, no figures. Revised to reference and describe a connection to Eisenstein's "periodic completion"of the Weierstrass zeta function
openalex created_date 2018/06/13 · arxiv created 2018/06/20 · openalex publication_date 2018/07/01 · arxiv updated 2019/06/03 · openalex updated_date 2026/08/06
A “modified” variant of the Weierstrass sigma, zeta, and elliptic functions is proposed whereby the zeta function is redefined by ζ(z) ↦ ζ̃(z) ≡ ζ(z)−γ2z, where γ2 is a lattice invariant related to the almost-holomorphic modular invariant of the quasi-modular-invariant weight-2 Eisenstein series. If ωi is a primitive half-period, ζ̃(ωi) = πωi*/A, where A is the area of the primitive cell of the lattice. The quasiperiodicity of the modified sigma function is much simpler than that of the original, and it becomes the building-block for the modular-invariant formulation of lowest-Landau-level wavefunctions on the torus. It is suggested that the “modified” sigma function is more natural than the original Weierstrass form, which was formulated before quasi-modular forms were understood. For the high-symmetry (square and hexagonal) lattices, the modified and original sigma functions coincide.