A real-valued function \(u(x,y)\) is harmonic if it satisfies Laplace's equation: \(\dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial y^2}=0\). If \(f=u+iv\) is analytic, both \(u\) and \(v\) are harmonic. The Cauchy-Riemann equations relate \(u\) and \(v\) but are not the definition of harmonicity.