  We propose an analytical approach for calculating the linear and nonlinear
optical (NLO) responses of monolayer transition metal dichalcogenides (TMDs)
including excitonic effects. An effective Hamiltonian reproducing the trigonal
warping (TW) of the energy dispersion, is used to derive analytical expressions
for excitonic matrix elements. Based on this approach, we provide an
informative diagram, which encompasses all excitonic selection rules. The
diagram enables us to identify main transitions for the first-, second- and
third-order optical responses. As a case study, we calculate the optical
conductivity and second/third harmonic generation responses of monolayer MoS2
and demonstrate that the analytical approach accurately reproduces the spectra
obtained using the Bethe-Salpeter equation (BSE). Therefore, the analytical
approach provides an efficient and reliable method for calculating the NLO
spectra as accurate as the full BSE method. Moreover, it enables us to obtain
valuable physical insight into the fundamental transitions responsible for
individual resonances, which is not straightforward in the full BSE method.
