= Green function on a Riemann surface
{c}
{title2=$G_R(q,p)$}
= Green functions on a Riemann surface
{c}
{synonym}
With the positive logarithmic convention, a Green function with pole $p$ on a noncompact <Riemann surface> is positive and harmonic away from $p$, has local form $-\log|\zeta|$ plus a harmonic function in a coordinate $\zeta(p)=0$, and is minimal among positive functions with these properties. It is obtained, when finite, as the increasing limit of zero-boundary Green functions on an exhaustion by compact bordered subdomains. On the <unit disk> it is
$$
G_{\mathbb D}(z,p)=\log\left|\frac{1-\overline pz}{z-p}\right|.
$$
Its sign convention is $-\Delta G=2\pi\delta_p$ in a conformal coordinate, with the opposite overall sign from conventions using $\log|\zeta|$ as the singular part.
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