An analytic function is a type of complex function that is differentiable at every point in its domain. This means that it 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 complex analysis, a branch of mathematics that studies functions of complex variables. One key property of analytic functions is that they are continuous and smooth, meaning they have no abrupt changes or breaks. They also satisfy the Cauchy-Riemann equations, which are conditions that relate the real and imaginary parts of the function. These properties make analytic functions useful in various fields, including physics and engineering.