  Long-time-integrated quantities in stochastic processes, in or out of
equilibrium, usually exhibit rare but huge fluctuations. Work or heat
production is such a quantity, of which the probability distribution function
displays an exponential decay characterized by the large deviation function
(LDF). The LDF is often deduced from the cumulant generating function through
the inverse Fourier transformation. The saddle-point integration method is a
powerful technique to obtain the asymptotic results in the Fourier integral,
but a special care should be taken when the saddle point is located near a
singularity of the integrand. In this paper, we present a modified saddle-point
method to handle such a difficulty efficiently. We investigate the dissipated
and injected heat production in equilibration processes with various initial
conditions, as an example, where the generating functions contain branch-cut
singularities as well as power-law ones. Exploiting the new modified
saddle-point integrations, we obtain the leading finite-time corrections for
the LDF's, which are confirmed by numerical results.
