Solution of Laplace's equation : Orthogonal set of functions

We shall now consider a method of solving the boundary value problem in electrostatics where one uses a basis of orthogonal functions. The basic idea is as follows. For a class of differential operators ( $\cal O$ ) defined over a region of space, there is a set of solutions

$${\cal O} \psi_n(x) = \epsilon_n \psi_n(x).$$

The solutions $\psi_n(x)$ are called the eigenfunctions and $\epsilon_n$ the eigenvalues. Here a point in the region of space is denoted by $x$. Further, the eigenfunctions $\psi$ are orthogonal, i.e. $\int dx \psi^*_m(x) \psi_n(x) = 0$ if $m \ne n$. Here it is understood that the integration is performed over the region of space over which the operator $\cal O$ is defined. By suitable scaling, we can normalise these functions. Thus, they satisfy orthonormality relation

$$\int dx \psi^*_m(x) \psi_n(x) = \delta_{m,n}.$$

In addition, these function form a complete set. That is, they satisfy the relation

$$\sum_n \psi_n(x) \psi^*_n(x') = \delta(x - x').$$

The fact that the set of functions $\psi_n(x)$ form a complete and orthonormal set over a region of space means that any function $F(x)$, defined in the region of the set of functions can be expressed in the form

$$F(x) = \sum_n A_n \psi_n(x).$$ (60)

The coefficients $A_n$ appear to be arbitrary but actually, they are defined uniquely by minimising $\int \left \vert F(x) - \sum_n A_n \psi(x) \right \vert^2$ with respect to the variation of $A_n$'s. The relation is

$$A_n = \int dx \psi^*_n(x) F(x)$$ (61)