a meromorphic function
/ meh-roh-MOR-fik /
A holomorphic function is the gold standard — complex-differentiable, no surprises. But many of the functions you actually care about blow up at isolated points: 1/z explodes at z = 0, tan z explodes where cos vanishes. A meromorphic function is the natural larger class that allows exactly this kind of well-controlled blow-up: holomorphic everywhere except at isolated points where it has poles, and nothing worse.
Precisely, on a Riemann surface X a meromorphic function is a map f: X -> C union {infinity} (the Riemann sphere) that is holomorphic away from a discrete set of points, at each of which it has a pole — meaning that near such a point p, in a local coordinate z with z(p) = 0, f looks like g(z)/z^k with g holomorphic and g(0) nonzero, for some positive integer k (the order of the pole). Equivalently 1/f is holomorphic and vanishes at p. Crucially, the bad points are isolated and the singularity is only polar: no essential singularities (like e^(1/z)) and no branch points are allowed. Viewing f as a map to the sphere, a pole is simply a point sent to infinity, so a meromorphic function IS a holomorphic map X -> Riemann sphere.
Why it matters: on a COMPACT Riemann surface this is the decisive concept. There are NO nonconstant holomorphic functions to the plane (a global holomorphic function on a compact surface is constant, by the maximum principle), so the interesting functions are exactly the meromorphic ones — and these form a field that encodes the surface completely. On the Riemann sphere the meromorphic functions are precisely the rational functions p(z)/q(z). An honest caveat: 'meromorphic' is strictly weaker than 'holomorphic'; it permits poles but forbids essential singularities, so e^(1/z) near 0 is neither holomorphic nor meromorphic there.
The Weierstrass function wp(z) attached to a lattice L is meromorphic on the torus C/L: it is holomorphic except for a double pole at each lattice point (one pole on the torus itself), where near 0 it behaves like 1/z^2. Together with its derivative it generates ALL meromorphic functions on that torus, which turn out to be exactly the rational functions of wp and wp'.
The Weierstrass wp-function: meromorphic on a torus, with one double pole, generating the whole function field.
Meromorphic is not the same as 'holomorphic with mild singularities' in general — it specifically allows poles only. Essential singularities (e^(1/z)) and branch points are excluded; a function with an essential singularity is not meromorphic there.