Analytic functions are complex functions that are differentiable at every point in their domain. This means they can be represented by a power series, which is an infinite sum of terms calculated from the function's derivatives. Analytic functions are important in various fields, including mathematics, physics, and engineering, as they often model real-world phenomena. One key property of analytic functions is that they are smooth and continuous, meaning they have no abrupt changes or breaks. Additionally, if a function is analytic in a region, it can be extended to a larger domain, making it a powerful tool for solving complex problems in complex analysis.