The charged $W$-boson propagator in an electromagnetic field in the generalized $R_{\xi}$-gauge

An exact solution of the equation for the $W$-boson propagator in the generalized $R_\xi$-gauge with consideration for the action of a crossed electromagnetic field has been found. The obtained general expression is consistent with the known result for the propagator in the absence of electromagnetic field, as well as with that in its presence in the cases of the unitary and Feynman gauges. The principle importance of the use of the $W$-boson propagator in the $R_{\xi}$-gauge in studying corresponding loop processes in electromagnetic field is discussed.