Weierstrass's Elliptic Functions

Invariants

In a deleted neighborhood of the origin, the Laurent series expansion of is

$\wp(z;\omega_1,\omega_2)=z^{-2}+\frac{1}{20}g_2z^2+\frac{1}{28}g_3z^4+O(z^6)$

where

and

The numbers g₂ and g₃ are known as the invariants. The summations after the coefficients 60 and 140 are the first two Eisenstein series, which are modular forms when considered as functions G₄(τ) and G₆(τ), respectively, of τ = ω₂/ω₁ with Im(τ) > 0.

Note that g₂ and g₃ are homogeneous functions of degree −4 and −6; that is,

and

Thus, by convention, one frequently writes and in terms of the period ratio and take to lie in the upper half-plane. Thus, and .

The Fourier series for and can be written in terms of the square of the nome as

and

where is the divisor function. This formula may be rewritten in terms of Lambert series.

The invariants may be expressed in terms of Jacobi's theta functions. This method is very convenient for numerical calculation: the theta functions converge very quickly. In the notation of Abramowitz and Stegun, but denoting the primitive half-periods by, the invariants satisfy

$g_2(\omega_1,\omega_2)= \frac{\pi^4}{12\omega_1^4} \left( \theta_2(0,q)^8-\theta_3(0,q)^4\theta_2(0,q)^4+\theta_3(0,q)^8 \right)$

and

$g_3(\omega_1,\omega_2)= \frac{\pi^6}{(2\omega_1)^6} \left[ \frac{8}{27}\left(\theta_2(0,q)^{12}+\theta_3(0,q)^{12}\right)\right.$

$\left. {} - \frac{4}{9}\left(\theta_2(0,q)^4+\theta_3(0,q)^4\right)\cdot \theta_2(0,q)^4\theta_3(0,q)^4 \right]$

where is the period ratio and is the nome.

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Weierstrass's Elliptic Functions - Invariants