6.2: Harmonic Functions
- Page ID
- 6502
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We start by defining harmonic functions and looking at some of their properties.
A function \(u(x, y)\) is called harmonic if it is twice continuously differentiable and satisfies the following partial differential equation:
\[\nabla ^2 u = u_{xx} + u_{yy} = 0. \label{6.2.1} \]
Equation \ref{6.2.1} is called Laplace’s equation. So a function is harmonic if it satisfies Laplace’s equation. The operator \(\nabla ^2\) is called the Laplacian and \(\nabla ^2 u\) is called the Laplacian of \(u\).