Holomorphicity couples real and imaginary parts by Cauchy--Riemann, forcing both to be harmonic.Suggest a slogan
Let \(\Omega\subseteq\CC\) be open and let \(f(z) = u(x, y) + iv(x, y)\) be holomorphic on \(\Omega\), with \(u,v\) real-valued and \(z=x+iy\). Then \(u\) and \(v\) are harmonic on \(\Omega\).
The derivative of a holomorphic function is holomorphic, so \(f\) is infinitely differentiable and \(u,v\) have continuous partial derivatives of all orders. By the Cauchy–Riemann equations \(u_x=v_y\) and \(u_y=-v_x\), so \(u_{xx}=v_{yx}=v_{xy}=-u_{yy}\) and \(v_{xx}=-u_{yx}=-u_{xy}=-v_{yy}\), using equality of mixed partials. Hence \(\Delta u=\Delta v=0\).