Definition 27.2.1 (Zero).label Let $E$ be a complete separated locally convex space over $\complex$, $U \subset \complex$ be open, $f \in H(U; E)$, and $z_{0} \in U$, then $z_{0}$ is a zero of $f$ of multiplicity $n \in \natp$ if there exists $g \in H(U; E)$ such that $f(z) = (z - a)^{n} g(z)$ for all $z \in \bracs{g \ne 0}$.