 Solution

Here $L_3=-i\hslash (x_1{\partial }_{x_2}-x_2{\partial }_{x_1})$. The radial factor and $x_3^n$ are annihilated by the angular derivative, whileThe product rule therefore givesReplacing $i$ by $-i$ gives eigenvalue $-m\hslash$ for $(x_1-i{x}_2{)}^m{x}_3^{n}f(r)$.