Harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R {\displaystyle f:U\to \mathbb {R} } , where U {\displaystyle U} is an open subset of R n {\displaystyle \mathbb {R} ^{n}} , that satisfies Laplace's equation, that is, ∂ 2 f ∂ x 1 2 + ∂ 2 f ∂ x 2 2 + ⋯ + ∂ 2 f ∂ x n 2 = 0 {\displaystyle {\frac {\partial ^{2}f}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}+\cdots +{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}=0} everywhere on U {\displaystyle U} . This is usually written as ∇ 2 f = 0 {\displaystyle \nabla ^{2}f=0} or Δ f = 0 {\displaystyle \Delta f=0} == Etymology of the term "harmonic" == The descriptor "harmonic" in the name "harmonic function" originates from a point on a taut string which is undergoing harmonic motion.