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.

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.

scholarly journals On a Conjecture of Alzer, Berg, and Koumandos

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1031
Author(s):  
Ladislav Matejíčka
Keyword(s):  
Gamma Function ◽  
Open Problem ◽  
Logarithmic Derivative ◽  
Completely Monotonic

In this paper, we find a solution of an open problem posed by Alzer, Berg, and Koumandos: determine ( α , m ) ∈ R + × N such that the function x α | ψ ( m ) ( x ) | is completely monotonic on ( 0 , ∞ ) , where ψ ( x ) denotes the logarithmic derivative of Euler’s gamma function.

scholarly journals A logarithmically completely monotonic function involving the gamma function and originating from the Catalan numbers and function

2015 ◽  
Vol 3 (4) ◽  
pp. 140 ◽  
Author(s):  
Fang-Fang Liu ◽  
Xiao-Ting Shi ◽  
Feng Qi
Keyword(s):  
Gamma Function ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Catalan Numbers ◽  
Monotonic Function ◽  
Completely Monotonic Function ◽  
Completely Monotonic ◽  
Logarithmically Completely Monotonic Function ◽  
And Function

<p><span>In the paper, the authors find necessary conditions and sufficient conditions for a function involving the gamma function and originating from investigation of properties of the Catalan numbers and function in combinatorics to be logarithmically completely monotonic.</span></p>

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