- Title
- On The Accuracy Of Asymptotic Approximations To The Log-Gamma And Riemann-Siegel Theta Functions
- Creator
- Brent, Richard P.
- Relation
- ARC.DP140101417 http://purl.org/au-research/grants/arc/DP140101417
- Relation
- Journal Of The Australian Mathematical Society Vol. 107, Issue 3, p. 319-337
- Publisher Link
- http://dx.doi.org/10.1017/S1446788718000393
- Publisher
- Cambridge University Press
- Resource Type
- journal article
- Date
- 2019
- Description
- We give bounds on the error in the asymptotic approximation of the log-Gamma function lnΓ(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 |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.
- Subject
- asymptotics; gamma function; log-gamma function; riemann zeta function; riemann–siegel theta function
- Identifier
- http://hdl.handle.net/1959.13/1452407
- Identifier
- uon:44432
- Identifier
- ISSN:1446-7887
- Language
- eng
- Reviewed
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