Better than Plemelj: An integration technique for calculating the generalized plasma dispersion function with applications to Thomson scattering

Kinetic plasma studies, such as nonlinear plasma instability analysis and Thomson scattering spectrum calculations, often require computing integrals of a velocity distribution function over a set of complex poles. Previous methods have invoked the Plemelj theorem and struggled with higher order poles, convergence, computational inefficiency, and numerical inaccuracy. We forgo the Plemelj contour approach and have developed a novel, efficient, accurate, and simple to implement direct integration method that can perform the calculation for an arbitrary number of arbitrary order poles. Example calculations are provided for multiple discretization schemes (e.g., finite difference or discontinuous Galerkin methods) to work with a variety of numerical plasma simulation techniques (e.g., Vlasov, PIC, Monte Carlo). We use our method to develop a forward model for calculating the Thomson scattering spectra of arbitrary non-relativistic non-Maxwellian collisonal magnetized plasmas. In addition, higher order pole calculations are explored and used to study the nonlinear mode coupling of an upper hybrid wave to an ion acoustic wave.