Some definations

Laplace’s equation

In n dimensions, the problem is to find twice-differentiable real-valued functions, \varphi of real variables, x_i, x_2, \cdots, x_n‘, such that

{\partial^2 \varphi \over \partial x_1^2 } +{\partial^2 \varphi \over \partial x_2^2 } +\cdots+{\partial^2 \varphi \over \partial x_n^2 } = 0.

This is often written as

\nabla^2 \varphi = 0


\Delta \varphi = 0.

Harmonic function

A harmonic function is a twice continuously differentiable function f : \mathbf{U} \rightarrow \mathbf{R} (where U is an open subset of Rn) which satisfies Laplace’s equation.