Uniform Asymptotic Expansions for Saddle Point Integrals-Application to a Probability Distribution Occurring in Noise Theory

The noncentral chi-square distribution occurs in noise interference problems. When the number of degrees of freedom becomes large, the middle portion of the distribution is given by the central limit theorem, and the tails by a classical saddle point expansion. Here recent work by N. Bleistein and F. Ursell on “uniform” asymptotic expansions is combined and extended to obtain an asymptotic series which apparently holds over the entire range of the distribution. General methods for expanding saddle point integrals in uniform asymptotic series are discussed. Recurrence relations are given for the coefficients in two typical cases, (i) when there are two saddle points and (ii) when there is only one saddle point but it lies near a pole or a branch point.