Definition:Analytic Function/Complex Plane

From ProofWiki
Jump to navigation Jump to search


Let $U \subset \C$ be an open set.

Let $f : U \to \C$ be a complex function.

Then $f$ is analytic in $U$ if and only if for every $z_0 \in U$ there exists a sequence $\sequence {a_n}: \N \to \C$ such that the series:

$\ds \sum_{n \mathop = 0}^\infty a_n \paren {z - z_0}^n$

converges to $\map f z$ in a neighborhood of $z_0$ in $U$.

Also known as

An analytic complex function is also referred to as a holomorphic function.

Some sources use the term regular function, but this is not endorsed on $\mathsf{Pr} \infty \mathsf{fWiki}$ as the word regular is overused.

The order of words can be varied: complex analytic function.

Also see

  • Results about analytic complex functions can be found here.