The theory of elliptic complexes of linear partial differential operators is closely interwoven with complex analysis. In particular, the Dolbeault complex is at the same time an important example of an elliptic complex and a tool for investigating more general ones. This thesis is mainly concerned with integral representations. The formula of Martinelli-Bochner provides one of the simplest integral representations for holomorphic functions. In this research I am concerned with elliptic systems, both determined and overdetermined. They admit, locally, left fundamental solutions. Green's integrals associated to them are natural analogues of the Martinelli-Bochner integral of complex analysis. In this dissertation I apply Green's integrals to the Cauchy problem for elliptic systems and to the question of the validity of the Poincaré lemma for elliptic differential complexes.