Sigma Functions

entire transcendental functions introduced by K. Weierstrass in his theory of elliptic functions. The principal sigma function (there are four in all) is

where w = 2mω₁+ 2nω₂ (ω₁ and ω₂ are two numbers such that ω₁/ω₂ is not real) and m and n independently run through all positive and negative integers, other than m = n = 0. The function σ(z) has simple zeros when z = w— that is, at the vertices of the parallelograms forming a regular lattice in the z-plane. These parallelograms are obtained from the parallelogram with vertices at the points 0, 2ω₁, 2ω₂, and 2(ω₁ + ω₂) by means of translations along its sides.

The functionσ(z) can be used to determine the Weierstrass zeta function ζ(z) and the Weierstrass elliptic function ℘(z):

Let ω₃ = – ω₁ – ω₂ and ζ(ω_k) = η_K, where k = 1, 2, 3. The formulas

σ(z + 2w_k) = – σ(z) exp [2η_k(z + w_k)] k = 1, 2, 3

express the property of quasi periodicity of the function σ(z). The equations

define the three remaining sigma functions. We have σ(0) = 0 and σ_k(0) = 1, k = 1, 2, 3. The function σ(z) is odd, and the other sigma functions are even.

Any elliptic function f(z) with periods 2ω₁, and 2ω₂ can be rationally expressed in terms of sigma functions by means of the formula

where C is a constant and a₁ …, a, and b₁, …, b_r are, respectively, complete systems of zeros and poles of f(z) satisfying the condition a₁ + … + a_r = b₁ + … + b_r. Sigma functions are closely related to theta functions.