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As conjectured by T. Przebinda, the transfer of characters in the Howe'scorrespondence should be obtained via the Cauchy-Harish-Chandra integral. Inthis paper, we prove that the conjecture holds for the dual pair $(G = U(p, q),G' = U(r, s))$, $p+q = r+s$, starting with a discrete series representation$\Pi$ of $\widetilde{U}(p, q)$.

Transfer of characters for discrete series representations of the unitary groups in the equal rank case via the Cauchy-Harish-Chandra integral