A harmonic function is a twice continuously differentiable function that satisfies Laplace's equation, meaning its second partial derivatives sum to zero. This property implies that harmonic functions are smooth and exhibit many interesting characteristics, such as being infinitely differentiable and conforming to the maximum principle, which states that a harmonic function achieves its maximum and minimum values on the boundary of a domain. These functions are crucial in various applications, including potential theory and the study of the Dirichlet problem.
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