= Dirichlet Green function between concentric spheres
{c}
{title2=$\nabla^2G=-4\pi\delta^{(3)}$}
For the gap $a<r<b$, put $u_\ell(r)=r^\ell-a^{2\ell+1}r^{-\ell-1}$, $v_\ell(r)=r^{-\ell-1}-r^\ell/b^{2\ell+1}$ and $d_\ell=(2\ell+1)[1-(a/b)^{2\ell+1}]$. With orthonormal <spherical harmonics>, the <Dirichlet Green function> is
$$
G(\mathbf r,\mathbf r')=4\pi\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}
\overline{Y_{\ell m}(\Omega)}Y_{\ell m}(\Omega')\frac{u_\ell(r_<)v_\ell(r_>)}{d_\ell}.
$$
For a short verification, $u_\ell(a)=v_\ell(b)=0$ and both solve the radial <Laplace equation>. Their <Wronskian> is $u_\ell v_\ell'-u_\ell'v_\ell=-d_\ell/r^2$. Thus the radial <derivative> jump is $-1/r'^2$; completeness of the <spherical harmonics> gives the stated <Dirac delta function> normalization. A <point charge> has <electric potential> $qG/(4\pi\epsilon_0)$, and the inner-wall <surface charge density> is $-q\partial_rG|_{a^+}/(4\pi)$, with the angular source coordinates held fixed.
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