Solutions to Laplace's Equation:   $\nabla^2 V = 0$

2D Cartesian:	$$\nabla^2 V \equiv \DbyD{^2 V}{x^2} + \DbyD{^2 V}{y^2} = 0$$ $$V(x,y) = \left. x \atop 1 \right\} \left. y \atop 1 \right\ . . . . . . t\} \quad \hbox{\rm + permutations } (x \leftrightarrow y) .$$
3D Cartesian:	$$\nabla^2 V \equiv \DbyD{^2 V}{x^2} + \DbyD{^2 V}{y^2} + \DbyD{^2 V}{z^2} = 0$$ $$V(x,y,z) = \left. x \atop 1 \right\} \left. y \atop 1 \ri . . . . . . sqrt{p^2 - q^2} \; z \atop \sin \sqrt{p^2 - q^2} \; z \right\}$$ $$\hbox{\rm + all permutations } \{x,y,z\} .$$
2D Plane Polar:	$$\nabla^2 V \equiv {1 \over r} \DbyD{}{r} \left( r \DbyD{V}{r} \right) + {1 \over r^2} \DbyD{^2 V}{\theta^2} = 0$$ $$V(r,\theta) = \left. \ln r \atop 1 \right\} + \left. r^n \a . . . . . . \right\} \left. \cos n \theta \atop \sin n \theta \right\}$$
3D Cylindrical:	$$\nabla^2 V \equiv {1 \over \rho} \DbyD{}{\rho} \left( \rho . . . . . . + {1 \over \rho^2} \DbyD{^2 V}{\phi^2} + \DbyD{^2 V}{z^2} = 0$$ $$V(\rho,\phi,z) = \left. J_n(k \rho) \atop N_n(k \rho) \righ . . . . . . op \sin n \phi \right\} \left. e^{kz} \atop e^{-kz} \right\}$$ where $J_n(k\rho) \to$ Bessel functions   and  $N_n(k\rho) \to$ Neumann functions.
3D Spherical:	$$\nabla^2 V \equiv {1 \over r^2} \DbyD{}{r} \left( r^2 \DbyD{V . . . . . . ight) + {1 \over r^2 \sin^2 \theta} \DbyD{^2 V}{\phi^2} = 0$$ $$V(r,\theta,\phi) = \left. r^\ell \atop r^{-(\ell+1)} \right . . . . . . heta) \right\} \left. \cos m \phi \atop \sin m \phi \right\}$$ where $P^m_\ell(\cos \theta)$ are associated Legendre polynomials and $Q^m_\ell(\cos \theta)$ are associated Legendre polynomials of the second kind. If axial symmetry  then ${\displaystyle V(r,\theta,\phi) = \left. r^\ell \atop r^{-(\ell+1)} \right\} \left. P_\ell(\cos \theta) \atop Q_\ell(\cos \theta) \right\} }$ where $P_\ell(\cos \theta)$ are Legendre polynomials and $Q_\ell(\cos \theta)$ are Legendre polynomials of the second kind.

Match linear combinations of the forms above to the appropriate boundary conditions imposed by (e.g.) conducting surfaces (equipotentials) and any requirements that $V \goestoas{r \to \infty} 0$ etc.