Global harmonic conjugate criterion by vanishing periods (${\oint }_{\gamma }(u_x-i{u}_y)dz=0$)

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 $2\pi i$.

For $|z|>1$, both logarithms in the display can use their principal logarithms. The real part is $\log (|z+1{|}^2/|z-1{|}^2)$ and the derivative is $2/(z+1)-2/(z-1)$. The two residues cancel on a loop surrounding both points. Equivalently $f=4{\sum }_{m\ge 0}{z}^{-(2m+1)}/(2m+1)$ is globally single-valued outside the unit circle, although individual logarithms of $z+1$ and $z-1$ are not.

// Destination: complex-dynamics.bigb