In real analysis and complex analysis, branches of mathematics, the identity theorem for analytic functions states: given functions f and g analytic on a domain D), if f = g has an accumulation point in D, then f = g on D.

Thus an analytic function is completely determined by its values on a single open neighborhood in D, or even a countable subset of D with an accumulation point (provided this contains a converging sequence together with its limit). This is not true in general for real-differentiable functions, even infinitely real-differentiable functions. In comparison, analytic functions are a much more rigid notion.

The underpinning fact from which the theorem is established is the expandability of a holomorphic function into its Taylor series.

The connectedness assumption on the domain D is necessary. For example, if D consists of two disjoint open sets on another.

Lemma

on a domain D.

, since both functions are analytic.

be the smallest nonnegative integer:

.

, the fiber.

Proof

have the same Taylor expansion

is nonempty, open, and closed. Then by connectedness.

is nonempty.

have non-zero radius of convergence is open.

is an intersection of closed sets, so it's closed.

Full characterisation

Since the identity theorem is concerned with the equality of two holomorphic functions. The following result can be found in.

Claim

denote a non-empty, connected open subset of the complex plane the following are equivalent.

1. ;

2. contains an accumulation point;

3. .

Proof

2) holds trivially.

3) is shown in section Lemma in part with Taylor expansion at accumulation point, just substitute g=0.

1) is shown in section Proof with set where all derivatives of f-g vanishes, just substitute g=0.

Q.E.D.

See also

• Analytic continuation

• Identity theorem for Riemann surfaces

References