Order of a zero
Definition. If \(f\) is holomorphic near \(a\) and not identically zero, then \(a\) is a zero of order \(m\) when \(f(z)=(z-a)^m g(z)\) with \(g\) holomorphic and \(g(a)\ne0\). The order is the first nonzero Taylor coefficient.
Isolated zeros
Theorem. Zeros of a nonzero holomorphic function are isolated. If zeros accumulated inside the domain, the Taylor expansion at the accumulation point would have all coefficients zero, forcing the function to vanish nearby.
Identity theorem
Theorem. If two holomorphic functions on a connected domain agree on a set with an accumulation point in the domain, then they agree everywhere. Apply isolated zeros to their difference.
Analytic continuation uniqueness
Corollary. A holomorphic continuation, if it exists along a connected overlap, is unique. Local agreement propagates across overlapping discs by the identity theorem.
Multiplicity and products
Proposition. Orders add under multiplication: \(\operatorname{ord}_a(fg)=\operatorname{ord}_a f+\operatorname{ord}_a g\), with poles treated later as negative orders. This arithmetic of orders underlies the argument principle.
Real-variable contrast
Remark. A smooth real function can vanish on an interval and then become nonzero. A holomorphic function cannot do this unless it is globally zero on the connected component.
Local factorization and multiplicity
Theorem. If \(f\) is holomorphic near \(a\), not identically zero, and \[ f(a)=f'(a)=\cdots=f^{(m-1)}(a)=0,\qquad f^{(m)}(a)\ne0, \] then \[ f(z)=(z-a)^mg(z) \] for a holomorphic \(g\) with \(g(a)\ne0\). The integer \(m\) is the order of the zero.
Proof. Factor the first nonzero term from the Taylor series: \[ f(z)=(z-a)^m \sum_{k\ge0}\frac{f^{(m+k)}(a)}{(m+k)!}(z-a)^k. \] The bracket is holomorphic and nonzero at \(a\).
Identity theorem
Theorem. If holomorphic functions \(f,g\) on a domain agree on a set with an accumulation point in the domain, then \(f\equiv g\).
Proof. Apply local factorization to \(h=f-g\). Unless \(h\) vanishes identically near the accumulation point, its zeros there are isolated, contradicting accumulation. The set on which \(h\) vanishes in a neighborhood is both open and closed in the connected domain, so it is all of \(\Omega\).
Orders under algebraic operations
Writing \(\operatorname{ord}_a f=m\), local factorization gives \[ \operatorname{ord}_a(fg) =\operatorname{ord}_af+\operatorname{ord}_ag, \] and, when \(g\not\equiv0\), \[ \operatorname{ord}_a(f/g) =\operatorname{ord}_af-\operatorname{ord}_ag. \] Negative orders describe poles. If orders of \(f\) and \(g\) differ, then \(\operatorname{ord}_a(f+g)\) is their minimum; equal orders can cancel and raise the result.
Hurwitz stability of zeros
Theorem (Hurwitz). If nonvanishing holomorphic \(f_n\) converge locally uniformly to \(f\) on a domain, then either \(f\) is nonvanishing or \(f\equiv0\).
Proof. If \(f\not\equiv0\) has a zero at \(a\), choose a small circle around \(a\) on which \(f\ne0\). Uniform convergence and Rouché's theorem imply \(f_n\) and \(f\) have the same positive number of zeros inside for all large \(n\), a contradiction.
Zeros are thus stable under locally uniform perturbation when counted with multiplicity, provided none cross the boundary of the region being counted.