A function is called "holomorphic" if it is complex differentiable at every point in a given domain. This means that the function can be represented by a power series in that region, making it smooth and continuous. Holomorphic functions are essential in complex analysis, a branch of mathematics that studies functions of complex variables. One important property of holomorphic functions is that they are infinitely differentiable, which means you can take derivatives of any order. Additionally, holomorphic functions obey the Cauchy-Riemann equations, which provide a condition for a function to be holomorphic. These functions also exhibit unique behaviors, such as conformality, which preserves angles locally.