Consider the function $f : \mathbb{R}^3 - (0,0,\mathbb{R} ) \to \mathbb{C}$
$$ f(x,y,z) = \frac{r}{x + i y} = \frac{\sqrt{x^2 + y^2 + z^2}}{x + i y}. $$
This function is harmonic, satisfying $$ (\partial_x^2 + \partial_y^2 + \partial_z^2 )f = 0.$$
We are told that harmonic functions can be decomposed with spherical harmonics with $r^\ell Y^m_\ell(\theta, \phi)$, but because this function is not defined on the line $x = y = 0$, this will not work. What then, is the representation theory of this function? Notice that if one defines the infinitesimal rotation generators \begin{align} L_1 &= x \partial_y - y \partial_x \\\\ L_2 &= y \partial_z - z \partial_y \\\\ L_3 &= z \partial_x - x \partial_z \end{align} then one can compute the action of the $SO(3)$ casimir on this function is $$ (L_1^2 + L_2^2 + L_3^2 )f = 0 $$ which should only be satisfied if $f = const$. However, $f$ is clearly not a constant! So what is the $SO(3)$ representation theory of this harmonic function?