A harmonic function is a type of mathematical function that satisfies Laplace's equation, meaning it is twice continuously differentiable and its second derivatives sum to zero. These functions are often used in various fields such as physics, engineering, and mathematics to describe phenomena like heat distribution and fluid flow. Harmonic functions have the property of being smooth and having no local maxima or minima within a given domain, except at the boundaries. They are closely related to concepts like potential theory and complex analysis, where they play a crucial role in understanding the behavior of analytic functions.