Elliptic function-field decomposition ($\mathbb{C}(\mathbb{C}/\Lambda )=\mathbb{C}(\wp )\oplus {\wp }^{\prime }\mathbb{C}(\wp )$)

Every elliptic function has a unique expression $A(\wp )+B(\wp ){\wp }^{\prime }$ with rational functions $A,B$. Its even part is rational in the Weierstrass elliptic function; its odd part divided by ${\wp }^{\prime }$ is even and meromorphic, so is also rational in $\wp$. The local even Laurent series at a half-period and at the pole justify meromorphic descent through the double cover of the Riemann sphere.