Identity theorem for Riemann surfaces
Encyclopedia
In mathematics
, the identity theorem for Riemann surface
s is a theorem
that states that a holomorphic function
is completely determined by its values on any subset of its domain that has a limit point
.
and 
be Riemann surfaces, let X be connected, and let 
be holomorphic. Suppose that 
for some subset 
that has a limit point, where 
denotes the restriction
of
to 
. Then 
(on the whole of 
).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the identity theorem for Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s is a theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
that states that a holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
is completely determined by its values on any subset of its domain that has a limit point
Limit point
In mathematics, a limit point of a set S in a topological space X is a point x in 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 contains a point of S other than x itself. Note that x does not have to be an element of S...
.
Statement of the theorem
Let and be Riemann surfaces, let X be connected, and let be holomorphic. Suppose that for some subset that has a limit point, where denotes the restrictionRestriction
Restriction may refer to:* Restriction , an aspect of a mathematical function* Restrictions , an album by Cactus* Restriction enzyme, a type of enzyme that cleaves genetic material...
of
to . Then (on the whole of ).