Computing the incomplete gamma function to arbitrary precision

Abstract

I consider an arbitrary-precision computation of the incomplete Gamma function from the Legendre continued fraction. Using the method of generating functions, I compute the convergence rate of the continued fraction and find a direct estimate of the necessary number of terms. This allows to compare the performance of the continued fraction and of the power series methods. As an application, I show that the incomplete Gamma function Γ(a,z) can be computed to P digits in at most O(P) long multiplications uniformly in z for Rez > 0. The error function of the real argument, erf x, requires at most O(P2/3) long multiplications. © Springer-Verlag Berlin Heidelberg 2003.