Harmonic functions are a class of smooth functions that satisfy the Laplace equation, which is a partial differential equation (PDE) defined as the sum of the second partial derivatives equaling zero. In other words, harmonic functions are solutions to the Laplace equation.
Harmonic functions play a crucial role in many areas of mathematics, including complex analysis, potential theory, and the study of Green’s functions. They also have applications in physics and engineering, particularly in the study of heat transfer, fluid dynamics, and electromagnetism. The properties of harmonic functions, such as their mean value property and maximum principle, make them useful for solving boundary-value problems.