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Identity TheoremEdit

The Identity Theorem is a cornerstone of complex analysis, reflecting the rigid nature of holomorphic objects. It asserts that a pair of holomorphic functions that agree on a set with a limit point inside a connected region must in fact agree everywhere on that region. In practical terms, this means a holomorphic function is determined by its values on a relatively small subset, provided that subset accumulates somewhere inside the domain. This rigidity underpins the idea of analytic continuation, where knowing a function on a small area lets us extend its behavior in a principled way to larger regions.

Formulated in the language of domains and analytic functions, the Identity Theorem ties local behavior (power series expansions around points) to global consequences. It is one of the clearest illustrations in complex analysis of how local analyticity controls global structure. The theorem also frames a basic dichotomy: either a holomorphic function vanishes only at isolated points, or it vanishes everywhere on a connected domain.

Identity Theorem

Formal statement

Let D be a connected open subset of the complex plane, and let f and g be holomorphic function on D. If the set { z ∈ D : f(z) = g(z) } has a limit point in D, then f ≡ g on D. Equivalently, if h(z) = f(z) − g(z) is holomorphic on D and h vanishes on a set with an accumulation point in D, then h ≡ 0 on D.

As a corollary, if a holomorphic function f on D vanishes on a set with a limit point in D, then f ≡ 0 on D. This leads to the familiar fact that the zeros of a nonzero holomorphic function are isolated.

Consequences

  • Uniqueness of analytic continuation: if two analytic continuations of a function agree on a region that has an accumulation point, they agree everywhere on their common domain analytic continuation.
  • Zeros of a nonzero holomorphic function are discrete: a holomorphic function cannot have a dense or accumulating zero set inside a connected domain unless it is identically zero.
  • Global constraints from local data: the values of a holomorphic function on a small, suitably dense subset determine its values on the entire connected domain.

Examples

  • Let f be holomorphic on a domain D and suppose f(z) = 0 for all z in a sequence {z_n} ⊂ D with z_n → z0 ∈ D. By the Identity Theorem, if {z_n} has a limit point in D, then f ≡ 0 on D.
  • If two holomorphic functions f and g agree on the integers—an infinite set with a limit point in many domains—then, under appropriate domain conditions, f ≡ g on the connected region of holomorphy containing those integers.

Sketch of proof

Define h = f − g, which is holomorphic on D. Suppose h is not identically zero. Then h has a Taylor expansion about any point a ∈ D: h(z) = Σ_{n≥m} c_n (z − a)^n, with c_m ≠ 0. If a set of zeros {z_k} of h accumulates at a within D, then h(z_k) = 0 for all k forces all coefficients c_n to vanish, contradicting c_m ≠ 0. Therefore h must be identically zero. This is the core idea: the accumulation of zeros forces the entire function to vanish, reflecting the rigidity of holomorphic functions.

Generalizations

  • The theorem extends to holomorphic functions on connected open subsets of complex manifolds, with the same qualitative conclusion.
  • Real-analytic analogues exist: two real-analytic functions that agree on a set with a limit point in an interval agree on the interval, mirroring the complex case.
  • Related results connect to the principle of analytic continuation and to uniqueness theorems for solutions to complex differential equations.

Historical notes

The Identity Theorem crystallized in the development of complex analysis in the 19th century, as mathematicians explored how power-series representations govern global behavior. It is closely tied to the maturation of the concept of analytic continuation and to the broader study of the rigidity of analytic structures.

Applications

  • Determining when two holomorphic functions are the same based on their values on a small subset.
  • Establishing that analytic continuations are unique.
  • Supporting the study of zeros and factorization of holomorphic functions, via the dichotomy between isolated zeros and identically vanishing functions.

See also