On the Accuracy of Asymptotic Approximations to the Log-Gamma and Riemann-Siegel Theta Functions

Abstract

We give bounds on the error in the asymptotic approximation of the log-Gamma function ln Γ(z) for complex z in the right half-plane. These improve on earlier bounds by Behnke and Sommer [Theorie der analytischen Funktionen einer komplexen Ver ¨anderlichen, 2nd edn (Springer, Berlin, 1962)], Spira [‘Calculation of the Gamma function by Stirling’s formula’, Math. Comp. 25 (1971), 317–322], and Hare [‘Computing the principal branch of log-Gamma’, J. Algorithms 25 (1997), 221–236]. We show that |Rk+1(z)/Tk(z)| < √ πk for nonzero z in the right half-plane, where Tk(z) is the kth term in the asymptotic series, and Rk+1(z) is the error incurred in truncating the series after k terms. We deduce similar bounds for asymptotic approximation of the Riemann–Siegel theta function ϑ(t). We show that the accuracy of a well-known approximation to ϑ(t) can be improved by including an exponentially small term in the approximation. This improves the attainable accuracy for real t > 0 from O(exp(−πt)) to O(exp(−2πt)). We discuss a similar example due to Olver [‘Error bounds for asymptotic expansions, with an application to cylinder functions of large argument’, in: Asymptotic Solutions of Differential Equations and Their Applications (ed. C. H. Wilcox) (Wiley, New York, 1964), 16–18], and a connection with the Stokes phenomenon Show more