= Weierstrass addition formula
{c}
{title2=$\wp(z+w)=-\wp(z)-\wp(w)+\tfrac14[(\wp'(z)-\wp'(w))/(\wp(z)-\wp(w))]^2$}
For generic fixed $w$, regard the right side as an <elliptic function> of $z$. The apparent <pole> at zero cancels, giving limiting value $\wp(w)$. At $z=w$ the slope ratio is finite. At $z=-w$, its squared slope has the same double principal part as $\wp(z+w)$ and no <simple pole> term. These exhaust the possible <poles> because the <Weierstrass elliptic function> has degree two. The difference is therefore an entire <elliptic function>, hence constant by <Liouville theorem>, and its value at zero is zero. <Meromorphic> continuation extends the identity to the exceptional parameters.
Back to article page
