In the book, the following problem is considered. Let W be a real vector space with a symplectic product and let Sp(W) be the corresponding Lie group. Consider a dual pair (G,G’) in Sp(W) in the sense of R. Howe. There is the so called Cauchy Harish-Chandra integral, which maps test functions with compact support on the Lie algebra of G to functions defined on the set of regular elements of the Lie algebra of G’. The aim of the book is to study properties of functions in the image of the Cauchy Harish-Chandra integral. For pairs formed by general linear groups over R, C or H, it is proved that functions in the image are orbital integrals. For pairs formed by unitary groups having the same rank, it is shown that they behave locally as orbital integrals and the author also proves suitable jump relations.