Show that if \(u,v\) are harmonic conjugates, then
- \(u^2-v^2\) is harmonic
- \(uv\) is harmonic.
- \(u_x\) is harmonic.
Solution
Write \(f=u+iv\), which is analytic.
\(f^2\) is analytic, and \(f^2 = (u+iv)^2 = u^2 - v^2 + i (2uv)\), which necessarily has harmonic components.
The imaginary part of \(f^2\) is \(2uv\), so \(uv\) is harmonic.
\(f'\) is analytic and \(f' = u_x + iv_x\), which has harmonic components.
Alternatively, \(uv\) is in the kernel of the Laplacian: \[ \laplacian(uv) &= (uv)_{xx} + (uv)_{yy} \\ &= (u_{xx}v + uv_{xx} + 2u_x v_x) + (u_{yy}v + uv_{yy} + 2u_y v_y) \\ &= (u_{xx} + u_{yy}) v + (v_{xx} + v_{yy}) u + 2(u_xv_x + u_yv_y) \\ &= (u_{xx} + u_{yy}) v + (v_{xx} + v_{yy}) u + 2(-u_x u_y + u_y u_x) && v_x = -u_y,\, v_y = u_x \\ &= (u_{xx} + u_{yy}) v + (v_{xx} + v_{yy})u \\ &= \laplacian(u)v + \laplacian(v)u \\ &= 0 .\]