= Solution
For a source in the exterior, its inverse point $\mathbf p$ lies strictly inside the excluded disc. The <method of images> construction of the <bounded exterior disk Green function> therefore allows an opposite logarithmic source there without adding a singularity to the domain. The candidate <Dirichlet Green function> is
$$
\boxed{G(\mathbf r;\mathbf r')=\frac1{2\pi}\left(\log|\mathbf r-\mathbf r'|-\log|\mathbf r-\mathbf p|-\log|\mathbf r'|\right).}
$$
Its <Laplacian> in the exterior is exactly $\delta(\mathbf r-\mathbf r')$, since the image singularity is outside that domain. The identity in (d) makes $G=0$ on the unit circle. Moreover
$$
\lim_{|\mathbf r|\to\infty}G(\mathbf r;\mathbf r')=-\frac1{2\pi}\log|\mathbf r'|,
$$
so it obeys the required boundedness condition. These checks establish the stated logarithm-of-a-ratio expression with $A=1/(2\pi)$. Bounded exterior <Dirichlet uniqueness for the Poisson equation> follows by <inversion in a circle> and removal of the bounded singularity at the origin.
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