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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.

The function field $k(X)$ of an irreducible algebraic variety consists of its rational functions.

At every point of a smooth algebraic curve, the local ring is a discrete valuation ring. Its valuation measures the order of vanishing of a rational function at that point.

The order of vanishing of a nonzero rational function $f$ at a point $p$ is the exponent of a local parameter in its local factorization. A negative order is the order of a pole.