A uniform asymptotic expansion for the incomplete gamma function

摘要

We describe a new uniform asymptotic expansion for the incomplete gamma function Γ(a,z) valid for large values of z. This expansion contains a complementary error function of an argument measuring transition across the point z=a (which is different from that in the well-known uniform expansion for large a of Temme), with easily computable coefficients that do not involve a removable singularity at z=a. Our expansion is, however, valid in a smaller domain of the parameters than that of Temme. Numerical examples are given to illustrate the accuracy of the expansion.