= Harmonicity under holomorphic composition
{title2=$\Delta(\psi\circ f)=|f'|^2(\Delta\psi)\circ f$}
If $f:U\to V$ is a <holomorphic function> between planar <open sets> and $\psi:V\to\mathbb R$ is a <harmonic function>, then $\psi\circ f$ is <harmonic>. In fact the <chain rule> and the <Cauchy-Riemann equations>, with $f=u+iv$, give
$$
\Delta(\psi\circ f)=\psi_{uu}|\nabla u|^2+2\psi_{uv}\nabla u\cdot\nabla v+\psi_{vv}|\nabla v|^2+\psi_u\Delta u+\psi_v\Delta v=|f'|^2(\Delta\psi)\circ f.
$$
Here the two <gradients> are <orthogonal> and have equal length, while $u,v$ are <harmonic>. The formula holds even where $f'=0$. A <biholomorphism> therefore induces inverse correspondences between <harmonic functions> by <function composition> with it and its inverse.
Back to article page
