Zeros and poles

In mathematics, a zero of a function f(x) is a value a such that f(a) = 0.

In complex analysis, zeros of holomorphic functions and meromorphic functions play a particularly important role because of the duality between zeros and poles.

A function f of a complex variable z is meromorphic in the neighbourhood of a point ${\displaystyle z_{0},}$ if either f or its reciprocal function 1/f is holomorphic in some neighbourhood of ${\displaystyle z_{0}}$ (that is, if f or 1/f is differentiable in a neighbourhood of ${\displaystyle z_{0}}$). If ${\displaystyle z_{0}}$ is zero of 1/f , then it is a pole of f.

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