Characterizations of boundaries of holomorphic 1-chains within ℂ ̂ × ℂ̂ and ℂ ̂ × ℂ̂

By way of a particular Cauchy-type integral, we characterize the closed rectifiable 1-currents γ, with support contained in ℂ ² and satisfying condition A ₁, that bound holomorphic 1-chains within ℂ̂× ℂ̂ . Also, we derive characterizations for the boundaries of holomorphic 1-chains within ℂ̂× ℂ̂, which yield examples of characterizations within a non-compact, non-Stein space. Additionally, we illustrate a connection between some of these characterizations and the Cauchy integral characterization of boundary values of meromorphic and holomorphic functions over a domain in ℂ.