A modified-residue prescription to calculate dynamical correlation functions

One of the challenges in using numerical methods to address many-body problems is the multi-dimensional integration over poles. More often that not, one needs such integrations to be evaluated as a function of an external variable. An example would be calculating dynamical correlations functions that are used to model response functions, where the external variable is the frequency. The standard numerical techniques rely on building an adaptive mesh, using special points in the Brillouin zone or using advanced smearing techniques. Most of these techniques, however, suffer when the grid is coarse. Here we propose that, if one knows the nature of the singularity in the integrand, one can define a residue and use it to faithfully estimate the integral and reproduce all the resulting singular features even with a coarse grid. We demonstrate the effectiveness of the method for different scenarios of calculating correlation functions with different resulting singular features, for calculating collective modes and densities of states. We also present a quantitative analysis of the error and show that this method can be widely applicable.