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


2D Cartesian:

     

\begin{displaymath}\nabla^2 V \equiv \DbyD{^2 V}{x^2} + \DbyD{^2 V}{y^2} = 0
\end{displaymath}


\begin{displaymath}V(x,y)
= \left. x \atop 1 \right\} \left. y \atop 1 \right\ . . . 
 . . . t\}
\quad \hbox{\rm + permutations } (x \leftrightarrow y) .
\end{displaymath}

3D Cartesian:

     

\begin{displaymath}\nabla^2 V \equiv
\DbyD{^2 V}{x^2} + \DbyD{^2 V}{y^2} + \DbyD{^2 V}{z^2} = 0
\end{displaymath}


\begin{displaymath}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\}
\end{displaymath}


\begin{displaymath}\hbox{\rm + all permutations } \{x,y,z\} .
\end{displaymath}

2D Plane Polar:

     

\begin{displaymath}\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
\end{displaymath}


\begin{displaymath}V(r,\theta)
= \left. \ln r \atop 1 \right\} + \left. r^n \a . . . 
 . . .  \right\}
\left. \cos n \theta \atop \sin n \theta \right\}
\end{displaymath}

3D Cylindrical:

     

\begin{displaymath}\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
\end{displaymath}


\begin{displaymath}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\}
\end{displaymath}

where $J_n(k\rho) \to$ Bessel functions   and  $N_n(k\rho) \to$ Neumann functions.
3D Spherical:

     

\begin{displaymath}\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
\end{displaymath}


\begin{displaymath}V(r,\theta,\phi)
= \left. r^\ell \atop r^{-(\ell+1)} \right . . . 
 . . . heta) \right\}
\left. \cos m \phi \atop \sin m \phi \right\}
\end{displaymath}

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.
Jess H. Brewer
2005-12-24