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ON THE ACCURACY OF ASYMPTOTIC APPROXIMATIONS TO THE LOG-GAMMA AND RIEMANN–SIEGEL THETA FUNCTIONS
Part of:
Zeta and $L$-functions: analytic theory
Elementary classical functions
Numerical approximation and computational geometry
Published online by Cambridge University Press: 21 December 2018
Abstract
We give bounds on the error in the asymptotic approximation of the log-Gamma function 
$\ln \unicode[STIX]{x1D6E4}(z)$ for complex 
$z$ in the right half-plane. These improve on earlier bounds by Behnke and Sommer [Theorie der analytischen Funktionen einer komplexen Veränderlichen, 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. Algorithms25 (1997), 221–236]. We show that 
$|R_{k+1}(z)/T_{k}(z)|<\sqrt{\unicode[STIX]{x1D70B}k}$ for nonzero 
$z$ in the right half-plane, where 
$T_{k}(z)$ is the 
$k$th term in the asymptotic series, and 
$R_{k+1}(z)$ is the error incurred in truncating the series after 
$k$ terms. We deduce similar bounds for asymptotic approximation of the Riemann–Siegel theta function 
$\unicode[STIX]{x1D717}(t)$. We show that the accuracy of a well-known approximation to 
$\unicode[STIX]{x1D717}(t)$ can be improved by including an exponentially small term in the approximation. This improves the attainable accuracy for real 
$t>0$ from 
$O(\exp (-\unicode[STIX]{x1D70B}t))$ to 
$O(\exp (-2\unicode[STIX]{x1D70B}t))$. 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.
Keywords
- Type
- Research Article
- Information
- Journal of the Australian Mathematical Society , Volume 107 , Issue 3 , December 2019 , pp. 319 - 337
- Copyright
- © 2018 Australian Mathematical Publishing Association Inc.
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