Identity theorem

Identity Theorem

The identity theorem is a fundamental result in complex analysis, a branch of mathematics that studies functions of complex variables. This theorem has significant implications in the theory of holomorphic functions and is instrumental in understanding the behavior of these functions.

Statement of the Identity Theorem

The identity theorem states that if two holomorphic functions \( f \) and \( g \) are defined on a connected open subset \( D \) of the complex plane \( \mathbb{C} \) and they agree on a set that has an accumulation point in \( D \), then \( f \) and \( g \) are identical on \( D \). Formally, if \( f \) and \( g \) are holomorphic on \( D \) and there exists a set \( S \subset D \) with an accumulation point in \( D \) such that \( f(z) = g(z) \) for all \( z \in S \), then \( f(z) = g(z) \) for all \( z \in D \).

Proof of the Identity Theorem

The proof of the identity theorem relies on the properties of holomorphic functions and the concept of analytic continuation. Here is a detailed proof:

1. **Holomorphic Functions and Power Series Representation**: Holomorphic functions can be locally represented by a convergent power series. Let \( f \) and \( g \) be holomorphic on \( D \). For any point \( z_0 \in D \), there exist power series expansions:

  \[
  f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n \quad \text{and} \quad g(z) = \sum_{n=0}^{\infty} b_n (z - z_0)^n
  \]
  which converge in some neighborhood of \( z_0 \).

2. **Equality on a Set with an Accumulation Point**: Suppose \( f \) and \( g \) agree on a set \( S \) with an accumulation point \( z_1 \in D \). This means \( f(z) = g(z) \) for all \( z \in S \).

3. **Uniqueness of Power Series Coefficients**: Since \( f \) and \( g \) agree on \( S \), their power series expansions around \( z_1 \) must be identical. Therefore, the coefficients \( a_n \) and \( b_n \) must be equal for all \( n \).

4. **Analytic Continuation**: The equality of the power series coefficients implies that \( f \) and \( g \) are equal in a neighborhood of \( z_1 \). By the principle of analytic continuation, if two holomorphic functions agree on any open subset of \( D \), they must agree on the entire connected domain \( D \).

Thus, \( f(z) = g(z) \) for all \( z \in D \).

Applications of the Identity Theorem

The identity theorem has several important applications in complex analysis and other areas of mathematics:

**Uniqueness of Analytic Continuation**: The theorem ensures that the analytic continuation of a holomorphic function is unique. If a function is extended analytically beyond its original domain, the extension is uniquely determined by the original function.

**Zeros of Holomorphic Functions**: If a holomorphic function has a set of zeros with an accumulation point, the function must be identically zero on the connected domain. This is a direct consequence of the identity theorem.

**Complex Dynamics**: In the study of dynamical systems in the complex plane, the identity theorem helps in understanding the behavior of iterated holomorphic functions and their fixed points.

**Functional Equations**: The identity theorem is used to solve functional equations involving holomorphic functions. If two solutions agree on a set with an accumulation point, they must be identical.

Related Concepts

**Analytic Continuation**: The process of extending the domain of a given analytic function beyond its original domain.
**Holomorphic Function**: A complex function that is differentiable at every point in its domain.
**Accumulation Point**: A point in the complex plane where every neighborhood contains at least one point from a given set distinct from the point itself.
**Power Series**: An infinite series of the form \( \sum_{n=0}^{\infty} a_n (z - z_0)^n \) representing a holomorphic function in a neighborhood of \( z_0 \).