Multiple-Order Singularity Expansion Method

Physical systems and signals are often characterized by complex functions of frequency in the harmonic-domain. The extension of such functions to the complex frequency plane has been a topic of growing interest as it was shown that specific complex frequencies could be used to describe both ordinary and exceptional physical properties. In particular, expansions and factorized forms of the harmonic-domain functions in terms of their poles and zeros under multiple physical considerations have been used. In this work, we start from a general property of continuity and differentiability of the complex functions to derive the multiple-order singularity expansion method. We rigorously derive the common singularity and zero expansion and factorization expressions, and generalize them to the case of singularities of arbitrary order, whilst deducing the behaviour of these complex frequencies from the simple hypothesis that we are dealing with physically realistic signals.