Zeros and poles

Summary

The zeros of a meromorphic function are the points where it takes the value $0$. Its poles are the points where it tends to $\infty$. See the definition. They are the two kinds of special points of a meromorphic function, and they are exchanged by $f \mapsto 1/f$. Counting them, each with its order, is the subject of the argument principle and of Rouché's theorem.