= Weierstrass elliptic differential equation
{c}
The <Weierstrass elliptic function> satisfies
$$
\wp'(z)^2=4\wp(z)^3-g_2\wp(z)-g_3.
$$
Writing $G_{2r}=\sum_{\omega\in\Lambda\setminus\{0\}}\omega^{-2r}$, the <Laurent coefficients of the Weierstrass elliptic function> give
$$
\wp(z)=z^{-2}+3G_4z^2+5G_6z^4+O(z^6),
\qquad
\wp'(z)=-2z^{-3}+6G_4z+20G_6z^3+O(z^5).
$$
Thus $g_2=60G_4$ and $g_3=140G_6$ cancel every nonremovable term in the <Laurent series> of $\wp'^2-4\wp^3+g_2\wp+g_3$ at each point of the <period lattice>. The resulting <elliptic function> is an <entire function> and is bounded, hence is zero by <Liouville theorem>.
Back to article page
