4.6 Laplace’s Equations

If \(f(z) = u(x,y) + iv(x,y)\) is analytic in a domain \(D\), then \(u\) and \(v\) satisfy Laplace’s equation \begin {align*} \nabla ^2 u & = u_{xx} + u_{yy} = 0\\ \nabla ^2v & = v_{xx} + v_{yy} = 0 \end {align*}

Solutions of Laplace’s equations having continuous second-order partial derivatives are called harmonic functions.