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A unified, fast, and effective approach is developed for numericalcalculation of the well-known plasma dispersion function with extensions fromMaxwellian distribution to almost arbitrary distribution functions, such as the$\delta$, flat top, triangular, $\kappa$ or Lorentzian, slowing down, andincomplete Maxwellian distributions. The singularity and analytic continuationproblems are also solved generally. Given that the usual conclusion$\gamma\propto\partial f_0/\partial v$ is only a rough approximation whendiscussing the distribution function effects on Landau damping, this approachprovides a useful tool for rigorous calculations of the linear wave andinstability properties of plasma for general distribution functions. Theresults are also verified via a linear initial value simulation approach.Intuitive visualizations of the generalized plasma dispersion function are alsoprovided.

Generalized Plasma Dispersion Function: One-Solve-All Treatment, Visualizations, and Application to Landau Damping