Holomorphic function

Definition

Definition for one-variable function

Let $U$ be an open subset (not necessarily connected, though we can for practical purposes restrict ourselves to domains -- connected open subsets) of $\mathbb {C}$. A function $f:U\to \mathbb {C}$ is termed a holomorphic function if it satisfies the following equivalent conditions:

1. $f$ is complex-differentiable at every point of $U$
2. $f$ is complex-differentiable at every point of $U$, and the function $f':U\to \mathbb {C}$ we obtain as the derivative, is a continuous function.
3. $f$ is infinitely often complex-differentiable. In other words, we can take the $n^{th}$ derivative of $f$ for any $n$
4. For any point $z_{0}\in U$, and any disc centered at $z_{0}$ of radius $r$, that lies completely inside $U$, $f$ can be expressed using a power series in $(z-z_{0})$
5. $f$ is a continuous function and satisfies the Cauchy-Riemann differential equations at every point in $U$
6. $f$ is a continuous function and its integral along the boundary of any triangle contained completely inside $U$ is zero.

Equivalence of definitions

• For equivalence of (2), (3), (4), refer complex-analytic implies holomorphic and holomorphic implies complex-analytic
• Goursat's lemma and Morera's theorem establish the equivalence of (1), (2) and (6)
• Looman-Menchoff theorem establishes the equivalence with (5).

Definition on Riemann surfaces

Let $M$ be a Riemann surface. A holomorphic function on $M$ is a map from $M$ to $\mathbb {C}$ whose restriction to any coordinate chart is a holomorphic function in the usual sense.