[en] The transition amplitude for quasi-elastic transfer reactions between heavy ions, given by distorted-waves theory, is evaluated in closed form. By means of suitable approximations the transfer partial-wave amplitude is expressed in terms of the elastic S-matrix elements in the initial and final channels and of known functions defined by the Coulomb radial integrals of the transfer form factor. The partial-wave summation is performed by using the Poisson sum formula in conjunction with Fourier-Bessel transform techniques for evaluating integrals over localized functions in l-space. The resulting expressions cover uniformly the whole angular range including forward and backward directions. The general structure of the excitation functions at 0⁰ and 180⁰ is discussed, including possible backward-angle enhancements of the transfer cross section. A crucial step in the derivation is the introduction of a mean elastic S-matrix, whose dependence on the initial and final angular momenta allows to display in detail the magnetic quantum number and Q-value dependences of the smooth and oscillatory parts of the cross section. Applications and extensions of the present formalism will be given in subsequent papers. (orig.)