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 smoothness. Holomorphic functions are defined on open subsets of the complex plane, and they can be represented by power series. One of the key properties of holomorphic functions is that they satisfy the Cauchy-Riemann equations, which are conditions that relate the real and imaginary parts of the function. These functions are important in complex analysis and have applications in various fields, including physics and engineering, due to their unique characteristics and behaviors.