\[\phi\]
in the \[xy\]
plane is a potential function if it satisfies \[\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} =0\]
.
The set of potential functions forma a vector space, since if
\[\phi, \psi\]
are poteial functions1.
\[\frac{\partial^2 0}{\partial x^2} + \frac{\partial^2 0}{\partial y^2} =0\]
, where 0=0(xy) is the function which returns zero for aal \[x,y\]
2.
\[a \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + b \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = \frac{\partial^2 (a \phi + b \psi )}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} =0\]
.We can write
\[\phi =\sum_{n=0}^{\infty} \sum_{i=0}^n a_{i, n-i} x^i y^{n-i}\]
then substitute into Poisson's equation to give \[\sum_{n=0}^{\infty} \sum_{i=2}^n a_{i, n-i} i(i-1)x^{i-2} y^{n-i}+ \sum_{n=0}^{\infty} \sum_{i=0}^{n-2} a_{i, n-i} (n-i)(n-i-1) x^i y^{n-i-2} =0\]
.Reindexing gives
\[\begin{equation} \begin{aligned} & \sum_{n=0}^{\infty} \sum_{i=0}^{n-2} a_{i+2, n-i-2} (i+2)(i+1)x^{i} y^{n-i-2}+ \sum_{n=0}^{\infty} \sum_{i=0}^{n-2} a_{i, n-i} (n-i)(n-i-1) x^i y^{n-i-2} \\ &= \sum_{n=0}^{\infty} \sum_{i=0}^{n-2} ( a_{i+2, n-i-2} (i+2)(i+1) +a_{i, n-i} (n-i)(n-i-1)) x^i y^{n-i-2} =0 \end{aligned} \end{equation}\]
.Hence
\[a_{i+2, n-i-2} (i+2)(i+1) +a_{i, n-i} (n-i)(n-i-1)=0\]
.Put
\[i=1, i=2\]
to get \[a_{3,n-3} =- \frac{a_{1,n-1} (n-1)(n-2)}{3 \times 2}\]
and \[a_{4,n-4} =- \frac{a_{2,n-2} (n-2)(n-3)}{4 \times 3}\]
.These are independent recurrence relations which determine
\[\phi\]
so the dimension of the vector space is 2.