  In this paper we present a complete linear synchrobetatron coupling formalism
by studying the transfer matrix which describes linear horizontal and
longitudinal motions. With the technique established in the linear
horizontal-vertical coupling study [D. Sagan and D. Rubin, Phys. Rev. ST Accel.
Beams {\bf 2}, 074001 (1999)], we found a transformation to block diagonalize
the transfer matrix and decouple the betatron motion and the synchrotron
motion. By separating the usual dispersion term from the horizontal coordinate
first, we were able to obtain analytic expressions of the transformation under
reasonable approximations. We also obtained the perturbations to the betatron
tune and the Courant-Snyder functions. The closed orbit changes due to finite
energy gains at rf cavities and radiation energy losses were also studied by
the 5$\times$5 extended transfer matrix with the fifth column describing kicks
in the 4-dimension phase space.
