= Solution
A <Dirichlet Green function> satisfies $\Delta_{\mathbf x}G_D(\mathbf x,\mathbf y)=\delta^{(2)}(\mathbf x-\mathbf y)$ in the domain and $G_D=0$ when $\mathbf x$ lies on its boundary. For a source $\mathbf y\ne0$ in the <unit disc>, put its opposite image at $\mathbf y^*=\mathbf y/|\mathbf y|^2$. The <method of images> gives
$$
\boxed{G_D(\mathbf x,\mathbf y)
=\frac1{2\pi}\log\frac{|\mathbf x-\mathbf y|}{|\mathbf y|\,|\mathbf x-\mathbf y^*|}}.
$$
The image lies outside the <unit disc>, so its logarithm is a <harmonic function> inside. The original logarithm has precisely the unit source strength found in part (a). On $|\mathbf x|=1$,
$$
|\mathbf x-\mathbf y|^2
=1-2\mathbf x\cdot\mathbf y+|\mathbf y|^2
=|\mathbf y|^2|\mathbf x-\mathbf y^*|^2,
$$
so the boundary value is zero. Equivalently, in <complex coordinates>, the <Dirichlet Green function inside the unit disk> is
$$
G_D(z,\zeta)=\frac1{2\pi}\log\left|\frac{z-\zeta}{1-z\bar\zeta}\right|.
$$
This also covers a source at the origin, where $G_D(z,0)=(2\pi)^{-1}\log|z|$.
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