Novel Proofs For Fundamental Identities Of Weierstrass Sigma And Jacobi Theta Functions

TL;DR

The paper addresses proving fundamental identities for the Weierstrass sigma function $\sigma(z)$ and Jacobi theta functions $\vartheta_j(z,\tau)$ using a contour-integral framework based on the argument principle. It develops a general method: for an entire, two-periodic quasi-periodic function $\varphi$ with $\varphi(z+\lambda_j)=e^{\alpha_j z+\beta_j}\varphi(z)$, the zero-count in a period parallelogram satisfies $N_\varphi=\frac{\alpha_1\lambda_2-\alpha_2\lambda_1}{2\pi i}$, and a contradiction in zeros forces $\varphi\equiv0$. The method is then used to prove key elliptic-function identities, including the Weierstrass $3$-term identity, Weierstrass fundamental identity, Jacobi addition formulas, Jacobi fundamental identities, and mixed-type sigma identities, leveraging Legendre’s relation $\eta_1\omega_3-\eta_3\omega_1=\frac{\pi i}{2}$ and theta-quasi-periodicity. Overall, it provides an alternative, contour-based route to elliptic-function identities with potential applicability to other quasi-periodic systems.

Abstract

In this note, we describe a general procedure to prove functional equations involving quasi-periodic functions. We give novel proofs for fundamental identities of Weierstrass sigma and Jacobi theta functions. Our method is based on the argument principle rather than the classical approach relying on Liouville's theorem.

Theorems & Definitions (14)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Lemma 2.4
• Lemma 2.5
• Theorem 2.6
• proof