 Global harmonic conjugate criterion by vanishing periods

For a harmonic function $u$, the function $g=u_x-i{u}_y$ is holomorphic. It is the derivative of a single-valued holomorphic function exactly when its integrals around all closed curves vanish. A primitive then has real part $u$ after adjusting a real constant. Simply connected domains satisfy this condition; on an annulus $u=\log |z|$ fails because the derivative has period $2\pi i$.