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, allowing for smooth curves without any breaks or sharp corners. Holomorphic functions are defined on open subsets of the complex plane, which is a two-dimensional space where each point represents a complex number. One of the key properties of holomorphic functions is that they can be represented by power series, similar to how polynomial functions work. This makes them very useful in various fields of mathematics and physics, particularly in areas like complex analysis and theoretical physics. Holomorphic functions also satisfy the Cauchy-Riemann equations, which are essential for determining their differentiability.