Harmonic Functions

Functions that solve Laplace’s equation and also satisfy the mean value property

1D

${\partial }_{x}V=0$

$V=mx+b$

• General solution

2D

Has no local extrema. Intuitively, it’s a surface that minimizes surface area while still meeting the boundary conditions much like the line solution in 1D. A ball would also always roll off the edge without finding a spot to rest

${\partial }_{x}V+{\partial }_{y}V=0$

3D

$V(r,\theta )=\sum _{l=0}^{\infty }\left(A_l{r}^l+\frac{B_l}{r^{l+1}}\right)P_l(\cos \theta )$

• General solution to Laplace’s equation in spherical coordinates
• $P_l$ is the Legendre polynomial

$A_l=\frac{1}{2{\epsilon }_0{R}^{l-1}}{\int }_0^{\pi }{\sigma }_0(\theta )P_l(\cos \theta )\sin \theta \mathrm{d}\theta$

$V(s,\varphi )=a_0+b_0\ln s+\sum _{k=1}^{\infty }[s^k(a_k\cos k\varphi +b_k\sin k\varphi )+s^{-k}(c_k\cos k\varphi +d_k\sin k\varphi )]$

• General solution to Laplace’s equation in cylindrical coordinates