a meromorphic function
/ mer-oh-MOR-fik /
If a holomorphic function is one that is complex-differentiable everywhere on its domain — perfectly smooth, no trouble spots — a meromorphic function is one allowed exactly one kind of blemish: poles, and nothing worse. It is holomorphic everywhere except at a scattering of isolated points where it blows up like 1/(z - z_0) to some power. Rational functions, tan z, and the gamma function are all meromorphic.
Precisely, f is meromorphic on a domain D if there is a set of isolated points (its poles) such that f is holomorphic on the rest of D and has a pole — not a removable or essential singularity — at each of those points. The poles cannot pile up inside D (their being isolated forbids that), so on any bounded closed region there are only finitely many. The cleanest way to picture a meromorphic function is as a ratio: locally, and for many global examples, f = g/h with g and h holomorphic, where the zeros of h that are not cancelled by zeros of g become the poles of f.
Meromorphic functions form a beautifully closed system: you can add, subtract, multiply, and divide them (dividing by one that is not identically zero) and stay meromorphic, so they form a field. This is why they are the natural setting for the residue theorem, the argument principle, and partial-fraction expansions. On the Riemann sphere, a function meromorphic on the whole extended plane turns out to be exactly a rational function — a striking rigidity that has no real-variable counterpart.
f(z) = (z^2 + 1)/((z - 1)(z + 3)^2) is meromorphic on the whole plane: it is holomorphic everywhere except a simple pole at z = 1 and a double pole at z = -3, where the denominator vanishes. tan z = sin z / cos z is meromorphic too, with simple poles where cos z = 0, namely z = pi/2 + n pi.
Holomorphic except for isolated poles — that is exactly what 'meromorphic' permits.
Meromorphic forbids essential singularities — e^(1/z) is NOT meromorphic at 0; allowing only poles is what gives the class its tidy algebra and its rationality on the sphere.