An Application of the Residue Calculus: The Distribution of the Sum of Nonhomogeneous Gamma Variates

The calculus of residues is one of the many beautiful tools that comes out of the field of complex variables. The calculus of residues is applied, together with the inversion formula for characteristic functions, to compute the non-gamma probability density function for the sum of gamma variates with different shape parameters. The distribution of the sum of gamma variates is needed in problems in statistical inference, as well as stochastic processes. This derivation seems more elegant than previous methods for deriving the density function of such a sum. Furthermore, the numerical computation is straightforward, especially in any symbolic computer language.