Holomorphic function

Definition

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be a function. The function $f$ is holomorphic on $U$ if $f$ is complex differentiable at each point of $U$. Let $z_0 \in \mathbb{C}$. A function $f$ is holomorphic at $z_0$ if $f$ is holomorphic on some open disc with center $z_0$.

Remarks

Polynomials in $z$ are holomorphic on $\mathbb{C}$, and $z \mapsto 1/z$ is holomorphic on $\mathbb{C} \setminus \{0\}$. Sums, products, quotients with a nonzero denominator and compositions of holomorphic functions are holomorphic. Each holomorphic function is analytic, see holomorphic functions are analytic. For this reason, many authors use the words holomorphic and analytic for the same functions. A function that is holomorphic on all of $\mathbb{C}$ is an entire function.

Explanations

L1 human

A holomorphic function is a function of a complex variable that has a complex derivative at each point of an open set. This one condition has strong consequences: the function then has derivatives of all orders and is equal to its Taylor series near each point.