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 (open and connected subset of
has an accumulation point in D, then f = g on D.[1] Thus an analytic function is completely determined by its values on a single open neighborhood in D, or even a countable subset of D (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.
Informally, one sometimes summarizes the theorem by saying analytic functions are "hard" (as opposed to, say, continuous functions which are "soft").
[citation needed] 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 a domain D agree on a set S which has an accumulation point
be the smallest nonnegative integer with
By holomorphy, we have the following Taylor series representation in some open neighborhood U of
is non-zero in some small open disk
This lemma shows that for a complex number
is a discrete (and therefore countable) set, unless
is nonempty, open, and closed.
have non-zero radius of convergence.
, they have holomorphic derivatives, so all
Since the identity theorem is concerned with the equality of two holomorphic functions, we can simply consider the difference (which remains holomorphic) and can simply characterise when a holomorphic function is identically
denote a non-empty, connected open subset of the complex plane.
it suffices to prove that the non-empty subset,
, is clopen (since a topological space is connected if and only if it has no proper clopen subsets).
Since holomorphic functions are infinitely differentiable, i.e.
has a convergent Taylor-series expansion centered on
3), fix an accumulation point
We now prove directly by induction that
be strictly smaller than the convergence radius of the power series expansion of
manipulation of the power series expansion yields Note that, since
is smaller than radius of the power series, one can readily derive that the power series
, the expression in (1) yields By the boundedness of
Via induction the claim holds.