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The order of vanishing of a nonzero rational function $f$ at a point $p$ is the exponent of a local parameter in its local factorization. A negative order is the order of a pole.

The local multiplicity in zeros and poles is positive for a zero and negative for a pole; it is zero at a regular nonvanishing point.

A function has a pole of order $k$ at $a$ when $(z-a{)}^{k}f(z)$ extends holomorphically and is nonzero at $a$.

A pole is the negative-order case in the zeros and poles classification of a meromorphic function.

Zeros and poles describe the positive and negative local orders of a meromorphic function. Locally $f(z)=(z-a{)}^{m}g(z)$ with $g$ holomorphic and nonzero at $a$: $m>0$ is a zero, $m<0$ is a pole, and $m=0$ is a regular nonzero value. A simple pole has $m=-1$, and the order of vanishing records $m$.