ABSTRACT
IG [Φ (s)] = < G (s) , Φ (s) >ℜ[s]=σ = ˆ σ+j∞ σ−j∞
G (s)Φ (s) ds. (18.1)
The test function Φ (s) has derivatives of any order along such a contour line in the s plane, and tends to zero more rapidly than any power of |s|. In what follows to lighten the notation we will sometimes write < G (s) , Φ (s) >, meaning < G (s) , Φ (s) >ℜ[s]=σ. As proposed in [22] and [24], the Dirac-delta impulse may be generalized, leading to a
distribution that is a generalized function of a complex variable. The generalized impulse may be denoted ξ(s) being a function of the complex variable s. We may define such a generalized Dirac-delta impulse by writing
IG [Φ (s)] = < ξ (s) , Φ (s) >ℜ[s]=σ = ˆ σ+j∞ σ−j∞
ξ (s)Φ (s) ds =
{ jΦ(0), σ = 0 0, σ 6= 0. (18.2)
The following properties are generalizations of properties of the usual real-variable distributions, and can be proven similarly to the corresponding proofs of the well-known theory of generalized functions.
