Find a harmonic conjugate for \[ u(x, y) = x^3 - 3xy^2 -x -y .\]
A harmonic conjugate \(v\) of \(u\) on a simply connected domain is computed as follows:
- Start with \(u\)
- Take \(\dd{}{x}\) to get \(u_x\)
- Apply CR to get \(u_x = v_y\)
- Take \(\int \dy\) to get \(v\) up to an unknown function \(f(x)\).
- Take \(\dd{}{x}\) to get \(v_x\) which involves \(f_x\)
- Apply CR to set \(v_x = -u_y\) and solve for \(f_x\)
- Compute \(\int f_x \dx\) to obtain \(f(x)\).
The same steps as a diagram:
Solution
First, check that \(u\) is actually harmonic: \[ \laplacian u = \dd{}{x}(3x^2-3y^2-1) + \dd{}{y}(-6xy - 1) = 6x + (-6x) = 0 .\]
Integrate \(v_y=u_x\) with respect to \(y\), \[ v_y = u_x = 3x^2 - 3y^2 - 1 \implies v = \int u_x \dy = 3x^2y - y^3 - y + f_1(x) .\] Now differentiate \(v\) with respect to \(x\) and set \(v_x = -u_y\): \[ v_x = 6xy + (f_1)_x = -u_y = 6xy + 1\implies f_1 = x + c_1 .\] Thus \[ v(x, y) = 3x^2y - y^3 - y + x + c_1 .\]