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A nonconstant meromorphic function on the complex plane is simply periodic when its group of periods is generated by one nonzero complex number. It is distinct from a doubly periodic elliptic function, whose period lattice has two real-linearly independent generators. Single periodicity alone imposes no algebraic addition law.

A nonconstant meromorphic function on a compact Riemann surface with exactly one simple pole gives a degree-one holomorphic map to the sphere. A degree-one map is a biholomorphism. The surface therefore has genus zero and the function is a global rational coordinate.

For a meromorphic function viewed as a holomorphic map to the unit Riemann sphere, this is the length density of the pulled-back spherical metric. Its square is the pulled-back area density. At a pole use the coordinate $1/f$, which gives the same density. Some conventions omit the factor two and use a sphere of area $\pi$; all area normalizations must then change accordingly.

The truncated counting function counts each distinct $a$-point of a meromorphic function once, rather than with its local multiplicity. It uses the same logarithmic radial weights as the Nevanlinna integrated counting function, including an origin contribution $\log r$ when appropriate. The difference $N-\overline{N}$ measures integrated excess multiplicity.

At $z=h_i/2$, evenness and periodicity give $\wp (z+h_i)=\wp (z)$. The Weierstrass half-period translation formula therefore identifies $(\wp (h_i/2)-e_i{)}^2$ with $(e_i-e_j)(e_i-e_l)$. Differentiating the Weierstrass half-period translation formula proves the identity, interpreted as equality of meromorphic functions, including the apparent singularities of its quotients.