Spectral Representation of the Dirac Delta Function

The physical difference between a finite and an infinite system has a profound impact on the mathematical structure of the corresponding Green's functions.

For a finite system, exemplified by a finite string, the Green's function is singular only at its poles, the eigenvalues of the system. By contrast, for an infinite system, for example, an infinite string, the Green's function is singular along each point of a line segment, a branch cut in the complex $\lambda$ -plane.