The authors derive an explicit formula for the two-dimensional free-surface Green function in water of infinite depth. The method combines various complex valued exponential integrals and elementary functions. The representation derived by the authors can easily be extended to the full plane and allows the consideration of a complex impedance. The representation is next used to solve a two-dimensional infinite-depth water-wave problem, which is formulated in terms of a boundary integral equation. From the numerical point of view the authors employ the boundary element method in solving such a problem. Finally, a benchmark problem based on half-circle is presented in order to validate the computations.