Show that if \(f = u+iv\) is holomorphic then \(u, v\) are harmonic.
Solution
Idea: use Cauchy-Riemann, take further derivatives, and use equality of partials.
By CR, \[ u_x = v_y && u_y = -v_x .\]
Differentiate with respect to \(x\): \[ u_{xx} = v_{yx} && u_{yx} = -v_{xx} .\]
Differentiate with respect to \(y\): \[ u_{xy} = v_{yy} && u_{yy} = -v_{xy} .\]
Clairaut’s theorem: partials are equal, so \[ u_{xx} - v_{yx} = 0 \implies u_{xx} + u_{yy} = 0 \\ \\ v_{xx} + u_{yx} = 0 \implies v_{xx} + v_{yy} = 0 \\ \\ .\]