A holomorphic function is a complex function that is differentiable at every point in its domain. This means that not only does the function have a derivative, but it also behaves nicely in terms of continuity and limits. Holomorphic functions are defined on open subsets of the complex plane and are characterized by their ability to be represented by power series. One of the key properties of holomorphic functions is that they satisfy the Cauchy-Riemann equations, which relate the real and imaginary parts of the function. These functions are important in various fields, including mathematics, physics, and engineering, due to their smoothness and the powerful results derived from them, such as Cauchy's integral theorem.