Completeness via Green's function

What can one say about the value of $\frac{1}{2\pi i} \oint_C G_\lambda (x;\xi)~d\lambda$ on the left hand side? If one knows that the set of eigenfunctions $\{ u_n(x)\}$ forms a complete set, then Eq.(1.14) on page  tells us that

$$\frac{\delta (x-x')}{\rho (x)} = \sum^\infty_{k=1}u_k(x)\overline{u}_k (x')~.$$ (430)

This is a necessary and sufficient condition for completeness. Combining it with Eq.(4.29), one finds that

$$\boxed{ \oint_C G_\lambda (x;\xi)~d\lambda= \frac{\delta (x-x')}{\rho (x)} }~.$$ (431)

This is the new criterion for completeness:
Equation (4.31) holds if and only if the spectral representation of $G_\lambda$ is based on a complete set of eigenfunctions.
This means that, if upon evaluating the left hand side of Eq.(4.27) along an asymptotically infinite circular contour $C$ , one finds that Eq.(4.31) holds, then one in guaranteed that the set of eigenfunctions obtained from Eq.(4.28) forms a complete set. The example of a free string in the next section illustrates this computational criterion.