Series expansions for the distribution of noncentral indefinite quadratic forms in complex normal variables

A new series expansion is developed for the probability distribution function and the cumulative distribution function for indefinite noncentral Hermitian quadratic forms in complex normal random variables. The moment generating function is inverted by contour integration using the Residue theorem. The function is separated into two parts, one part, containing an essential singularity, is expanded by Laurent series and the other part is expanded by Taylor series. The series are combined for evaluating the residue of the complete function. Several different series can be obtained by modifications of the basic approach. The series are computationally efficient and normally fast converging. The convergence rate depends on the eigenvalues separation. Multiple eigenvalues are allowed, and can be used to approximately replace a close pair of eigenvalues.