  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.
