scholarly journals Exact Values of the Gamma Function from Stirling’s Formula

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1058
Author(s):  
Victor Kowalenko
Keyword(s):  
Complex Plane ◽  
Gamma Function ◽  
Asymptotic Series ◽  
Stirling’S Formula ◽  
Borel Summation ◽  
Standard Terms ◽  
Asymptotic Forms

In this work the complete version of Stirling’s formula, which is composed of the standard terms and an infinite asymptotic series, is used to obtain exact values of the logarithm of the gamma function over all branches of the complex plane. Exact values can only be obtained by regularization. Two methods are introduced: Borel summation and Mellin–Barnes (MB) regularization. The Borel-summed remainder is composed of an infinite convergent sum of exponential integrals and discontinuous logarithmic terms that emerge in specific sectors and on lines known as Stokes sectors and lines, while the MB-regularized remainders reduce to one complex MB integral with similar logarithmic terms. As a result that the domains of convergence overlap, two MB-regularized asymptotic forms can often be used to evaluate the logarithm of the gamma function. Though the Borel-summed remainder has to be truncated, it is found that both remainders when summed with (1) the truncated asymptotic series, (2) Stirling’s formula and (3) the logarithmic terms arising from the higher branches of the complex plane yield identical values for the logarithm of the gamma function. Where possible, they also agree with results from Mathematica.

THE BURNSIDE APPROXIMATION OF GAMMA FUNCTION

2010 ◽  
Vol 08 (03) ◽  
pp. 315-322 ◽  
Author(s):  
XIQUAN SHI ◽  
FENGSHAN LIU ◽  
HONGMIN QU
Keyword(s):  
Gamma Function ◽  
Asymptotic Series ◽  
Stirling’S Formula ◽  
Completely Monotonic

Different from the famous Stirling's formula [Formula: see text], Burnside presented another formula [Formula: see text]. In this paper, some estimations and a convergent asymptotic series of b(s) are obtained. At the same time, it is proved that both -b(s) and [Formula: see text] are completely monotonic on the interval (½, ∞).

scholarly journals ON THE ACCURACY OF ASYMPTOTIC APPROXIMATIONS TO THE LOG-GAMMA AND RIEMANN–SIEGEL THETA FUNCTIONS

2018 ◽  
Vol 107 (3) ◽  
pp. 319-337
Author(s):  
RICHARD P. BRENT
Keyword(s):  
New York ◽  
Half Plane ◽  
Gamma Function ◽  
Asymptotic Approximation ◽  
Asymptotic Series ◽  
Theta Functions ◽  
Stirling’S Formula ◽  
Stokes Phenomenon ◽  
Principal Branch ◽  
The Right

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.

Hyperasymptotics

1990 ◽  
Vol 430 (1880) ◽  
pp. 653-668 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Transition Point ◽  
Asymptotic Series ◽  
Stokes Phenomenon ◽  
Numerical Computations ◽  
Borel Summation ◽  
Nested Sequence ◽  
Exponentially Small

We develop a technique for systematically reducing the exponentially small (‘superasymptotic’) remainder of an asymptotic expansion truncated near its least term, for solutions of ordinary differential equations of Schrödinger type where one transition point dominates. This is achieved by repeatedly applying Borel summation to a resurgence formula discovered by Dingle, relating the late to the early terms of the original expansion. The improvements form a nested sequence of asymptotic series truncated at their least terms. Each such ‘hyperseries’ involves the terms of the original asymptotic series for the particular function being approximated, together with terminating integrals that are universal in form, and is half the length of its predecessor. The hyperasymptotic sequence is therefore finite, and leads to an ultimate approximation whose error is less than the square of the original superasymptotic remainder. The Stokes phenomenon is automatically and exactly incorporated into the scheme. Numerical computations confirm the efficacy of the technique.

scholarly journals Plummeting and blinking eigenvalues of the Robin Laplacian in a cuspidal domain

2019 ◽  
Vol 150 (6) ◽  
pp. 2871-2893 ◽  
Author(s):  
Sergei A. Nazarov ◽  
Nicolas Popoff ◽  
Jari Taskinen
Keyword(s):  
Dimension Reduction ◽  
Discrete Spectrum ◽  
Spectral Theory ◽  
Real Axis ◽  
Complex Plane ◽  
Lipschitz Domain ◽  
High Rate ◽  
The Real ◽  
Robin Laplacian ◽  
Asymptotic Forms

We consider the Robin Laplacian in the domains Ω and Ωε, ε > 0, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient a is large enough, the spectrum of the problem in Ω is known to be residual and to cover the whole complex plane, but on the contrary, the spectrum in the Lipschitz domain Ωε is discrete. However, our results reveal the strange behaviour of the discrete spectrum as the blunting parameter ε tends to 0: we construct asymptotic forms of the eigenvalues and detect families of ‘hardly movable’ and ‘plummeting’ ones. The first type of the eigenvalues do not leave a small neighbourhood of a point for any small ε > 0 while the second ones move at a high rate O(| ln ε|) downwards along the real axis ℝ to −∞. At the same time, any point λ ∈ ℝ is a ‘blinking eigenvalue’, i.e., it belongs to the spectrum of the problem in Ωε almost periodically in the | ln ε|-scale. Besides standard spectral theory, we use the techniques of dimension reduction and self-adjoint extensions to obtain these results.

scholarly journals Stirling’s Formula and Its Extension for the Gamma Function

2013 ◽  
Vol 120 (8) ◽  
pp. 737 ◽  
Author(s):  
Dorin Ervin Dutkay ◽  
Constantin P. Niculescu ◽  
Florin Popovici
Keyword(s):  
Gamma Function ◽  
Stirling’S Formula
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