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We propose a new computational method of the Loschmidt amplitude in a genericspin system on the basis of the complex semiclassical analysis on thespin-coherent state path integral. We demonstrate how the dynamical transitionsemerge in the time evolution of the Loschmidt amplitude for the infinite-rangetransverse Ising model with a longitudinal field, exposed by a quantum quenchof the transverse field $\Gamma$ from $\infty$ or $0$. For both initialconditions, we obtain the dynamical phase diagrams that show the presence orabsence of the dynamical transition in the plane of transverse field after aquantum quench and the longitudinal field. The results of semiclassicalanalysis are verified by numerical experiments. Experimental observation of ourfindings on the dynamical transition is also discussed.

Complex semiclassical analysis of the Loschmidt amplitude and dynamical quantum phase transitions