# Holomorphic Function
> [!NOTE] Holomorphic Function
> A [[Complex Number|complex-valued]] function is *holomorphic* if at every point of a region there exists a [[neighbourhood]] in which the function is complex [[Derivative|differentiable]].
The requirement for holomorphic functions to be complex differentiable within a neighbourhood of all points of a region means that they are *infinitely differentiable*. It also means they are always locally equal to its own [[Taylor series]] and are thus also complex [[Analytic Function|analytic]] functions.
As holomorphic functions and complex analytic functions are equivalent, it is common for holomorphic functions to simply be called analytic functions, despite analytic functions being a broader category that contains real functions as well.
As holomorphic functions are complex differentiable everywhere, they also satisfy the [[Cauchy-Riemann equations]].
## Properties
- If two functions $f$ and $g$ in a domain $D$ are holomorphic, then $f\pm g$, $fg$, and $f\circ g$ are also holomorphic. $f/g$ is only holomorphic if $g$ has *no zeroes* in $D$.
- The real and imaginary parts of a holomorphic function are [[Harmonic Function|harmonic]] functions.
- [[Cauchy's integral theorem]] states that the [[contour integral]] of holomorphic functions along a simple closed contour in a [[Simply Connected Space|simply connected]] domain is always zero.
- [[Cauchy's integral formula]] states that any function that is holomorphic on a disk can be completely determined using the values on the boundary of the disk.