Identity theorem

Two holomorphic functions on a connected domain that agree on a set with an interior accumulation point agree everywhere.

 Ancestors (5)

 Incoming links (52)

 Synonyms (1)

codex/identity-theorem-for-holomorphic-functions

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (2)

Identity theorem by

Wikipedia Bot  1

 Read the full article

Identity theorem by

Ciro Santilli  40 Updated 2025-07-16

Essentially, defining an holomorphic function on any open subset, no matter how small, also uniquely defines it everywhere.

This is basically why it makes sense to talk about analytic continuation at all.

One way to think about this is because the Taylor series matches the exact value of an holomorphic function no matter how large the difference from the starting point.

Therefore a holomorphic function basically only contains as much information as a countable sequence of numbers.

 Read the full article

  See all articles in the same topic Create my own version