In mathematics, the identity theorem for

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...

s is a

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...

that states that a holomorphic function is completely determined by its values on any subset of its domain that has a

limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...

Statement of the theorem

Let

X

and

Y

be Riemann surfaces, let

X

be connected, and let

f, g : X \to Y

be holomorphic. Suppose that

f, _ = g, _

for some subset

A \subseteq X

that has a limit point, where

f, _ : A \to Y

denotes the

restriction Restriction, restrict or restrictor may refer to: Science and technology * restrict, a keyword in the C programming language used in pointer declarations * Restriction enzyme, a type of enzyme that cleaves genetic material Mathematics and logi ...

f

A

. Then

f = g

(on the whole of

X

References

* {{DEFAULTSORT:Identity Theorem For Riemann Surfaces Theorems in complex analysis Riemann surfaces