= Global harmonic conjugate criterion by vanishing periods
{title2=$\oint_\gamma(u_x-iu_y)\,dz=0$}
For a <harmonic function> $u$, the function $g=u_x-iu_y$ 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 $2\pi i$.
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