Identity theorem for holomorphic functions

Depends on

• The locally zero locus of a holomorphic function is clopen
• The order of a zero is the exponent in its local holomorphic factorization
• Linearity, product, reciprocal, and quotient rules for complex derivatives
• Complex differentiability at a point implies continuity there
• Interior, closure, boundary, limit point, isolated point and dense subset of a metric space
• A complex domain is a nonempty connected open subset of $\mathbb C$
• Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets

Used by

• The ring of holomorphic functions on a complex domain is an integral domain Corollary
• A divergent Blaschke sum: no nonzero Hardy function has these zeros Counterexample
• Agreement accumulating only at the boundary does not force a holomorphic identity Counterexample
• Local degree of a nonconstant holomorphic map Definition
• Meromorphic differentials, orders and residues Definition
• The complex-time heat kernel on a proper sector Definition
• Addition and duplication for ℘ Example
• Riesz measure of a log modulus records the holomorphic zeros Example
• A holomorphic function on a domain in ℂ² vanishing on a set with an accumulation point vanishes identically False statement
• Meromorphic Jensen identity with a zero or pole at the centre Lemma
• Separately holomorphic functions vanishing on a real box are zero Lemma
• Spectral product from traces of powers Lemma
• The fibres of lambda are exactly the Gamma(2)-orbits Lemma
• The germ neighborhoods form a Hausdorff, second-countable Riemann-surface atlas Lemma
• Transformation laws and S₃-action of the modular lambda function Lemma
• Transformation laws of the Jacobi theta function Lemma
• A power series of finite radius has a singular point on its circle of convergence Theorem
• Boundary values and log-integrability of Nevanlinna-class functions Theorem
• Complex-time heat operators form a bounded holomorphic semigroup Theorem
• Degree of a proper holomorphic map of Riemann surfaces Theorem
• Local power-map normal form on Riemann surfaces Theorem
• Open mapping theorem for holomorphic functions Theorem
• Residue theorem on a compact Riemann surface Theorem
• The Beta-Gamma identity Theorem
• The complex Pythagorean identity by the identity theorem Theorem
• The level-one valence formula Theorem
• The local Second Main Theorem on a punctured disc Theorem
• The q-expansion principle at the cusp Theorem
• The terminal germ of a continuation along a fixed path is chain-independent Theorem
• The Weierstrass product for reciprocal Gamma Theorem
• The zero set of a Hardy function satisfies the Blaschke condition Theorem
• Vitali-Porter convergence theorem for holomorphic functions Theorem
• Zeros of a nonzero holomorphic function are isolated Theorem

Dependency tree · two levels

28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• J. Lebl, Guide to Cultivating Complex Analysis, Theorem 2.4.7 (standard reference, not scraped)
• B. V. Shabat, Introduction to Complex Analysis, Theorem 2.28 (standard reference, not scraped)