An analytic function is a type of mathematical function that is defined by a power series in a neighborhood of every point in its domain. This means that around any point where the function is defined, you can express it as an infinite sum of terms involving powers of the variable. Analytic functions are smooth and continuous, making them important in various fields of mathematics and physics. One key property of analytic functions is that they are differentiable, meaning they have a derivative at every point in their domain. This differentiability leads to many useful results, such as the ability to represent them locally as polynomials. Examples of analytic functions include polynomials, exponential functions, and trigonometric functions.