  We present a unified description of the excitonic properties of four
monolayer transition-metal dichalcogenides (TMDC's) using an equation of motion
method for deriving the Bethe-Salpeter equation in momentum space. Our method
is able to cope with both continuous and tight-binding Hamiltonians, and is
less computational demanding than the traditional first-principles approach. We
show that the role of the exchange energy is essential to obtain a good
description of the binding energy of the excitons. The exchange energy at the
$\Gamma-$point is also essential to obtain the correct position of the
C-exciton peak. Using our model we obtain a good agreement between the Rydberg
series measured for WS$_2$. We discuss how the absorption and the Rydberg
series depend on the doping. Choosing $r_0$ and the doping we obtain a good
qualitative agreement between the experimental absorption and our calculations
for WS$_2$. We also derive a semi-analytical version of Ellitot's formula for
TMDC's.
