# Exact Values of the Gamma Function from Stirling’s Formula

## Abstract

**:**

## 1. Introduction

## 2. Stirling’s Formula

**Lemma**

**1.**

**Proof.**

**Definition**

**1.**

**Definition**

**2.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 3. Numerical Analysis

## 4. Mellin–Barnes Regularization

**Theorem**

**2.**

**Proof.**

## 5. Further Numerical Analysis

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Wikipedia, the Free Encyclopedia, Stirling’s Approximation. Available online: http://en.wikipe\protect\discretionary{\char\hyphenchar\font}{}{}dia.org/wiki/Stir\protect\discretionary{\char\hyphenchar\font}{}{}lings_approxi\protect\discretionary{\char\hyphenchar\font}{}{}mation (accessed on 8 June 2020).
- Whittaker, E.T.; Watson, G.N. A Course of Modern Analysis, 4th ed.; Cambridge University Press: Cambridge, UK, 1973; p. 252. [Google Scholar]
- Segur, H.; Tanveer, S.; Levine, H. (Eds.) Asymptotics beyond all Orders; Plenum Press: New York, NY, USA, 1991. [Google Scholar]
- Kowalenko, V. Towards a theory of divergent series and its importance to asymptotics. In Recent Research Developments in Physics; Transworld Research Network: Trivandrum, India, 2001; Volume 2, pp. 17–68. [Google Scholar]
- Kowalenko, V. Exactification of the asymptotics for Bessel and Hankel functions. Appl. Math. Comput.
**2002**, 133, 487–518. [Google Scholar] [CrossRef] - Kowalenko, V. The Stokes Phenomenon, Borel Summation and Mellin-Barnes Regularisation; Bentham Ebooks: Sharjah, UAE, 2009; Available online: http://www.bentham.org (accessed on 18 June 2020).
- Kowalenko, V.; Frankel, N.E.; Glasser, M.L.; Taucher, T. Generalised Euler-Jacobi Inversion Formula and Asymptotics beyond All Orders; London Mathematical Society Lecture Note 214; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Wikipedia, the Free Encyclopedia, Hadamard Regularization. Available online: https://en.wikipe\protect\discretionary{\char\hyphenchar\font}{}{}dia.org/wiki/Ha\protect\discretionary{\char\hyphenchar\font}{}{}da\protect\discretionary{\char\hyphenchar\font}{}{}mard_re\protect\discretionary{\char\hyphenchar\font}{}{}gula\protect\discretionary{\char\hyphenchar\font}{}{}rization (accessed on 11 April 2020).
- Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions; Dover: New York, NY, USA, 1965. [Google Scholar]
- Gradshteyn, I.S.; Ryzhik, I.M.; Jeffrey, A. (Eds.) Table of Integrals, Series and Products, 5th ed.; Academic Press: London, OH, USA, 1994. [Google Scholar]
- Kowalenko, V. Exactification of Stirling’s approximation for the logarithm of the gamma function. arXiv
**2014**, arXiv:1404.2705. [Google Scholar] - Paris, R.B.; Kaminski, D. Asymptotics and Mellin-Barnes Integrals; Cambridge University Press: Cambridge, UK, 2001. [Google Scholar]
- Paris, R.B. Hadamard Expansions and Hyperasymptotic Evaluation—An Extension of the Method of Steepest Descents; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Kowalenko, V. Applications of the cosecant and related numbers. Acta Appl. Math.
**2011**, 114, 15–134. [Google Scholar] [CrossRef] - Dingle, R.B. Asymptotic Expansions: Their Derivation and Interpretation; Academic Press: London, UK, 1973. [Google Scholar]
- Kowalenko, V. Euler and Divergent Series. Eur. J. Pure Appl. Math.
**2011**, 4, 370–423. [Google Scholar] - Wolfram, S. Mathematica—A System for Doing Mathematics by Computer; Addison-Wesley: Reading, MA, USA, 1992. [Google Scholar]
- Berry, M.V. Uniform asymptotic smoothing of Stokes’s discontinuities. Proc. R. Soc. Lond. A
**1989**, 422, 7–21. [Google Scholar] - Stokes, G.G. On the discontinuity of arbitrary constants which appear in divergent developments. In Collected Mathematical and Physical Papers; Cambridge University Press: Cambridge, UK, 1904; Volume 4, pp. 77–109. [Google Scholar]
- Olver, F.W.J. On Stokes’ phenomenon and converging factors. In Asymptotic and Computational Analysis; Wong, R., Ed.; Marcel-Dekker: New York, NY, USA, 1990; pp. 329–355. [Google Scholar]
- Berry, M.V.; Howls, C.J. Hyperasymptotics for integrals with saddles. Proc. R. Soc. Lond. A
**1991**, 434, 657–675. [Google Scholar] - Paris, R.B.; Wood, A.D. Exponentially-improved asymptotics for the gamma function. J. Comp. Appl. Math.
**1992**, 41, 135–143. [Google Scholar] [CrossRef] - Berry, M.V.; Howls, C.J. Hyperasymptotics. Proc. R. Soc. Lond. A
**1990**, 430, 653–658. [Google Scholar] - Berry, M.V. Asymptotics, superasymptotics, hyperasymptotics. In Asymptotics beyond all Orders; Segur, H., Tanveer, S., Levine, H., Eds.; Plenum Press: New York, NY, USA, 1991; pp. 1–9. [Google Scholar]
- Paris, R.B.; Wood, A.D. Stokes phenomenon demystified. IMA Bull.
**1995**, 31, 21–28. [Google Scholar] - Kowalenko, V. Reply to Paris’s comments on exacitification for the logarithm of the gamma function. arXiv
**2014**, arXiv:1408.1881. [Google Scholar]

**Figure 1.**$\Re ln\mathsf{\Gamma}\left(z\right)$ as a function of $\theta $ between 0 and $\pi $ for fixed values of $\left|z\right|$.

**Figure 2.**$\Im ln\mathsf{\Gamma}\left(z\right)$ as a function of $\theta $ between 0 and $\pi $ for fixed values of $\left|z\right|$.

**Figure 3.**The conventional Stokes multiplier ${S}^{+}$ (blue) vs. the smoothed version (red) for $\left|z\right|=3$ as a function of $\theta $.

**Table 1.**$ln\mathsf{\Gamma}\left(3\right)$ via (35) for various values of the truncation parameter, N.

N | Quantity | Value |
---|---|---|

$F\left(3\right)$ | 0.66546925487494697026844282871193190148012386819465 | |

TS | 0.02777777777777777777777777777777777777777777777777 | |

2 | ${R}_{2}^{SS}\left(3\right)$ | −0.0000998520927794385973038298896926468609453577911 |

Total | 0.69314718055994530944891677660001703239695628818129 | |

TS | 0.02767489711934156378600823045267489711934156378600 | |

5 | ${R}_{5}^{SS}\left(3\right)$ | 3.468406207072280893260950592903359343689305700 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}\phantom{\rule{0.277778em}{0ex}}$ |

Total | 0.69314718055994530941723212145817656807550013436025 | |

TS | 0.02767792490305420773799675002229807193246527808017 | |

11 | ${R}_{11}^{SS}\left(3\right)$ | 5.889005342650445782024537787635919634947854418 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ |

Total | 0.69314718055994530941723212145817656807550013436025 | |

TS | 0.02767792637739909405287985684200177174299855050138 | |

20 | ${R}_{20}^{SS}\left(3\right)$ | −6.0499126756131034798323054398369045866792543896 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}\phantom{\rule{0.277778em}{0ex}}$ |

Total | 0.69314718055994530941723212145817656807550013436025 | |

TS | 41.2834736138079254966213754129139774958755379575621 | |

30 | ${R}_{30}^{SS}\left(3\right)$ | −41.255795688122927157472586120167732829280161691396 |

Total | 0.69314718055994530941723212145817656807550013436025 | |

TS | 6.0039864088710184849557428450939638222762809177 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{25}$ | |

50 | ${R}_{50}^{SS}\left(3\right)$ | −6.003986408871018484955742842326171253776447002 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{25}\phantom{\rule{0.277778em}{0ex}}$ |

Total | 0.69314718055994530941723212145817656807550013436025 | |

$ln\mathsf{\Gamma}\left(3\right)$ | 0.69314718055994530941723212145817656807550013436025 |

N | Quantity | Value |
---|---|---|

$F(1/10)$ | 1.73997257040229101538752631827936332290183806908929 | |

TS | 0.83333333333333333333333333333333333333333333333333 | |

2 | ${R}_{1}^{SS}(1/10)$ | −0.3205932520014175333654707340213015524648898035500 |

Total | 2.25271265173420681535538891759139510377028159887258 | |

TS | −1.94444444444444444444444444444444444444444444444444 | |

3 | ${R}_{3}^{SS}(1/10)$ | 2.457184525776359388926618212330380430142089389324197 |

Total | 2.252712651734205959869700086165299308599483016422822 | |

TS | −5874.96031746031746031746031746031746031746031746031 | |

5 | ${R}_{5}^{SS}(1/10)$ | 5875.473057541649375261942492788406592113174013182747 |

Total | 2.252712651734205959869701646368495118615533791559062 | |

TS | −2.94867419474845489858725152842799901623431035195 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{13}$ | |

9 | ${R}_{9}^{SS}(1/10)$ | 2.948674194748506172595384719922447233767119265136 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{13}$ |

Total | 2.25271265173420595986970164636849511861562722229495 | |

TS | −3.60868558918311609670918346035346984255501011055 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{31}$ | |

15 | ${R}_{15}^{SS}(1/10)$ | 3.60868558918311609670918346035352111656314330204 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{31}$ |

Total | 2.252712651734205959869701646368495118615627222294953 | |

$ln\mathsf{\Gamma}(1/10)$ | 2.252712651734205959869701646368495118615626380692264 |

**Table 3.**$ln\mathsf{\Gamma}\left(z\right)$ via (25) with $\left|z\right|\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1/10$ for various values of the truncation parameter, N, and $\mathrm{arg}z$.

N | $\mathit{\theta}$ | Quantity | Value |
---|---|---|---|

$F\left(z\right)$ | 1.75803888205251701300823152720 + 0.38158365834299627447460123156$\phantom{\rule{0.166667em}{0ex}}i$ | ||

TS | 0.72168783648703220563643597562 − 2.36111111111111111111111111111$\phantom{\rule{0.166667em}{0ex}}i$ | ||

3 | $-\pi /6$ | ${R}_{3}^{SS}\left(z\right)$ | −0.2230295240392980035338083054 + 2.52524252152237336263247340087$\phantom{\rule{0.166667em}{0ex}}i$ |

$S{D}_{0}^{SS}\left(z\right)$ | 0 | ||

Total | 2.25669719450025121511085919742 + 0.54571506875425852599596352132$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$ln\mathsf{\Gamma}\left(z\right)$ | 2.25669719450025121511085784624 + 0.54571506875425852599596430142$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$F\left(z\right)$ | 1.88648341970221940135996338478 + 1.03535606610194782214347998160$\phantom{\rule{0.166667em}{0ex}}i$ | ||

TS | 2.87562548020794239198561 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{13}$ − 7.0718880105443602759020497 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{12}\phantom{\rule{0.166667em}{0ex}}i$ | ||

9 | $-6\pi /13$ | ${R}_{9}^{SS}\left(z\right)$ | −2.8756254802079022596675 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{13}$ + 7.0718880105448298576076792 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{12}\phantom{\rule{0.166667em}{0ex}}i$ |

$S{D}_{0}^{SS}\left(z\right)$ | 0 | ||

Total | 2.28780660084741914752819484319 + 1.50493777173150666351075080995$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$ln\mathsf{\Gamma}\left(z\right)$ | 2.28780660084741914752819484319 + 1.50493777173150666351075080994$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$F\left(z\right)$ | 1.93811875120925961146815019100 + 1.18372127170949939121742184779$\phantom{\rule{0.166667em}{0ex}}i$ | ||

TS | −6.2877629092633776151775 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{10}$ + 1.3406164598876901506339999 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{10}\phantom{\rule{0.166667em}{0ex}}i$ | ||

8 | $-8\pi /15$ | ${R}_{8}^{SS}\left(z\right)$ | 6.28776290922350273134009 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{10}$ − 1.3406164598401660891969575 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{10}\phantom{\rule{0.166667em}{0ex}}i$ |

$S{D}_{1}^{SS,L}\left(z\right)$ | 0.76110557640259383178972540915 + 0.07527936657383153773373307240$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Total | 2.30047548923720786540859926314 + 1.73424125265375514384575434070$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$ln\mathsf{\Gamma}\left(z\right)$ | 2.30047548923720786540859926314 + 1.73424125265375514384575434070$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$F\left(z\right)$ | 2.33668492162243351553206801970 + 1.825916904516483614559881067115$\phantom{\rule{0.166667em}{0ex}}i$ | ||

TS | −42.600558891527544536579217000 + 64.60897639111337349406319763878$\phantom{\rule{0.166667em}{0ex}}i$ | ||

4 | $-15\pi /16$ | ${R}_{4}^{SS}\left(z\right)$ | 42.0897905773173511704765793260 − 64.54765565501832133510270286263$\phantom{\rule{0.166667em}{0ex}}i$ |

$S{D}_{1}^{SS,L}\left(z\right)$ | 0.54123306366541416118208181725 + 1.072657474660830843519039814447$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Total | 2.36714967107765431061151216215 + 2.959895115272366617039415657712$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$ln\mathsf{\Gamma}\left(z\right)$ | 2.36714967107765431061151216215 + 2.959895115272366617039415657712$\phantom{\rule{0.166667em}{0ex}}i$ |

**Table 4.**$ln\mathsf{\Gamma}\phantom{\rule{-0.166667em}{0ex}}\left(3exp(i\pi /2)\right)$ via (25) for various values of N.

N | Quantity | Value |
---|---|---|

$F(3exp(i\pi /2\left)\right)$ | −4.3427565915140719616112579569 − 0.4895612973931192354299251350522$\phantom{\rule{0.166667em}{0ex}}i$ | |

$S{D}_{0}^{SL}(3exp(i\pi /2))$ | 3.256206078642828367679816468 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | |

Combined | −4.3427565882578658829684295892 − 0.4895612973931192354299251350522$\phantom{\rule{0.166667em}{0ex}}i$ | |

TS | $0$ | |

1 | ${R}_{1}^{SL}(3exp(i\pi /2))$ | − 0.0278840894653691199321777792256$\phantom{\rule{0.166667em}{0ex}}i$ |

Total | −4.3427565882578658829684295892 − 0.5174453868584883553621029142779$\phantom{\rule{0.166667em}{0ex}}i$ | |

TS | $0$ − 0.0278842394252900781377131527007$\phantom{\rule{0.166667em}{0ex}}i$ | |

6 | ${R}_{6}^{SL}(3exp(i\pi /2))$ | $0$ − 1.8907874105339892863379255 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}\phantom{\rule{0.166667em}{0ex}}i$ |

Total | −4.3427565882578658829684295892 − 0.51744555572628341890753115113225$\phantom{\rule{0.166667em}{0ex}}i$ | |

TS | $0$ − 0.0278842563298976281594154202028$\phantom{\rule{0.166667em}{0ex}}i$ | |

9 | ${R}_{9}^{SL}(3exp(i\pi /2))$ | $0$ + 3.2562060786428283676798164 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}\phantom{\rule{0.166667em}{0ex}}i$ |

Total | −4.3427565882578658829684295892 − 0.51744555572628341890753115113225$\phantom{\rule{0.166667em}{0ex}}i$ | |

TS | $0$ − 0.0278842691899612112195938305035$\phantom{\rule{0.166667em}{0ex}}i$ | |

15 | ${R}_{15}^{SL}(3exp(i\pi /2))$ | $0$ + 1.0856797027741987814423624 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}\phantom{\rule{0.166667em}{0ex}}i$ |

Total | −4.3427565882578658829684295892 − 0.51744555572628341890753115113225$\phantom{\rule{0.166667em}{0ex}}i$ | |

TS | $0$ − 52.07235660935681329352406137393$\phantom{\rule{0.166667em}{0ex}}i$ | |

30 | ${R}_{30}^{SL}(3exp(i\pi /2))$ | $0$ + 52.044472351023649035874440121314$\phantom{\rule{0.166667em}{0ex}}i$ |

Total | ||

TS | $0$ − 6.4908409843349435181620453 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{25}\phantom{\rule{0.166667em}{0ex}}i$ | |

50 | ${R}_{50}^{SL}(3exp(i\pi /2))$ | $0$ + 6.4908409843349435181620453 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{25}\phantom{\rule{0.166667em}{0ex}}i$ |

Total | ||

$ln\mathsf{\Gamma}(3exp(i\pi /2\left)\right)$ | −4.3427565882578658829684295892 − 0.51744555572268341890753115113225$\phantom{\rule{0.166667em}{0ex}}i$ |

**Table 5.**$ln\mathsf{\Gamma}\phantom{\rule{-0.166667em}{0ex}}\left(3exp\left(i\right(1/2+\delta \left)\pi \right)\right)$ via (24) for various values of $\delta $.

$\mathit{\delta}$ | Method | Value |
---|---|---|

1/10 | LogGamma[z] | −5.1085546405054331385771175 − 2.43504864133618239587613036$\phantom{\rule{0.166667em}{0ex}}i$ |

$S{D}_{1}^{SS,U}\left(z\right)$ | 0.0000000146924137960847328 + 0.00000000724920978735477097$\phantom{\rule{0.166667em}{0ex}}i$ | |

1st AF | −5.1085546405054331385771175 − 2.43504864133618239587613036$\phantom{\rule{0.166667em}{0ex}}i$ | |

−1/10 | LogGamma[z] | −3.1156770612855851062960250 + 0.79152717486178700663566144$\phantom{\rule{0.166667em}{0ex}}i$ |

3rd AF | −3.1156770612855851062960250 + 0.79152717486178700663566144$\phantom{\rule{0.166667em}{0ex}}i$ | |

1/100 | LogGamma[z] | −4.4448078360199294879676721 − 0.68426539470619315579497619$\phantom{\rule{0.166667em}{0ex}}i$ |

$S{D}_{1}^{SS,U}\left(z\right)$ | 0.0000000054543808883397577 − 0.00000000366845661861183983$\phantom{\rule{0.166667em}{0ex}}i$ | |

1st AF | −4.4448078360199294879676721 − 0.68426539470619315579497619$\phantom{\rule{0.166667em}{0ex}}i$ | |

−1/100 | LogGamma[z] | −4.2360547825638102221663061 − 0.35681003461125834209091866$\phantom{\rule{0.166667em}{0ex}}i$ |

3rd AF | −4.2360547825638102221663061 − 0.35681003461125834209091866$\phantom{\rule{0.166667em}{0ex}}i$ | |

1/1000 | LogGamma[z] | −4.3531757575591613140088085 − 0.53385166100905755261595669$\phantom{\rule{0.166667em}{0ex}}i$ |

$S{D}_{1}^{SS,U}\left(z\right)$ | 0.0000000065016016472424544 − 0.00000000038545945628149871$\phantom{\rule{0.166667em}{0ex}}i$ | |

1st AF | −4.3531757575591613140088085 − 0.53385166100905755261595669$\phantom{\rule{0.166667em}{0ex}}i$ | |

−1/1000 | LogGamma[z] | −4.3322909095906129602545969 − 0.50110130347126170951651903$\phantom{\rule{0.166667em}{0ex}}i$ |

3rd AF | −4.3322909095906129602545969 − 0.50110130347126170951651903$\phantom{\rule{0.166667em}{0ex}}i$ | |

1/10,000 | LogGamma[z] | −4.3438006028809735966127763 − 0.51908338527968766540121412$\phantom{\rule{0.166667em}{0ex}}i$ |

$S{D}_{1}^{SS,U}\left(z\right)$ | 0.0000000065123040290213875 − 0.00000000003856476898298508$\phantom{\rule{0.166667em}{0ex}}i$ | |

1st AF | −4.3438006028809735966127763 − 0.51908338527968766540121412$\phantom{\rule{0.166667em}{0ex}}i$ | |

−1/10,000 | LogGamma[z] | −4.3417121085407199183370966 − 0.51580834470414165478538635$\phantom{\rule{0.166667em}{0ex}}i$ |

3rd AF | −4.3417121085407199183370966 − 0.51580834470414165478538635$\phantom{\rule{0.166667em}{0ex}}i$ | |

1/20,000 | LogGamma[z] | −4.3438006028809735966127763 − 0.51908338527968766540121412$\phantom{\rule{0.166667em}{0ex}}i$ |

$S{D}_{1}^{SS,U}\left(z\right)$ | 0.0000000065123851251757157 − 0.00000000001928245580002624$\phantom{\rule{0.166667em}{0ex}}i$ | |

1st AF | −4.3438006028809735966127763 − 0.51908338527968766540121412$\phantom{\rule{0.166667em}{0ex}}i$ | |

−1/20,000 | LogGamma[z] | −4.3422344065179726897501879 − 0.51662687288967352139359494$\phantom{\rule{0.166667em}{0ex}}i$ |

3rd AF | −4.3422344065179726897501879 − 0.51662687288967352139359494$\phantom{\rule{0.166667em}{0ex}}i$ |

**Table 6.**Values of $ln\mathsf{\Gamma}\left({z}^{3}\right)$ with $\left|z\right|=1/10$ and varying N and $\theta $ in the Mellin–Barnes (MB)-regularized forms.

$\mathit{\theta}$ | N | Quantity | Value |
---|---|---|---|

$F\left({z}^{3}\right)$ | 4.3666691849467394839681993920 + 0.39773534871318634708519397906$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$T{S}_{3}\left({z}^{3}\right)$ | 1.964244428861064224518267$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ − 1.9641265777308664665975342$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}\phantom{\rule{0.166667em}{0ex}}i$ | ||

$-\pi /12$ | 3 | MB Int (M1 = 0) | −1.964241888183123016922580$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ + 1.964126965801012078012431$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}\phantom{\rule{0.166667em}{0ex}}i$ |

${S}_{MB}(0,{z}^{3})$ | 0 | ||

Total via M1 | 6.9073471261543351713515623993 + 0.7858054943246012439632804975$\phantom{\rule{0.166667em}{0ex}}i$ | ||

MB Int (M2 = −1) | −1.964246960282777058252170$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ + 1.964127748979378940448405$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}\phantom{\rule{0.166667em}{0ex}}i$ | ||

${S}_{MB}(1,{z}^{3})$ | 5.0720996540413295899999675136 − 0.783178366862435974379475023745$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Total via M2 | 6.9073471261543351713515623992 + 0.78580549432460124396328049761$\phantom{\rule{0.166667em}{0ex}}i$ | ||

LogGamma[zcube] | 6.9073471261543351713515623992 + 0.78580549432460124396328049761$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$F\left({z}^{3}\right)$ | 4.3790700299188033385944250366 − 1.38001259938612058620816944744$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$T{S}_{4}\left({z}^{3}\right)$ | 3.03718072746107697324584$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{11}$ − 7.332351579151624817641834$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{11}\phantom{\rule{0.166667em}{0ex}}i$ | ||

$7\pi /24$ | 4 | MB Int (M1 = 0) | −3.03718072743578478216056$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{11}$ + 7.33235157913793379318864$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{11}\phantom{\rule{0.166667em}{0ex}}i$ |

${S}_{MB}(0,{z}^{3})$ | 0 | ||

Total via M1 | 6.9082891384461367831946353384 − 2.749115044705401162838355615704$\phantom{\rule{0.166667em}{0ex}}i$ | ||

MB Int (M2 = 1) | −3.03718072748649559827233$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{11}$ + 7.33235157914968575273952$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{11}\phantom{\rule{0.166667em}{0ex}}i$ | ||

${S}_{MB}(1,{z}^{3})$ | 5.0710816111765935339415417904 − 1.175195955088607468412668235518$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Total via M2 | 6.9082891384461367827408810781 − 2.749115044705401165409263040026$\phantom{\rule{0.166667em}{0ex}}i$ | ||

LogGamma[zcube] | 6.9082891384461367827408810777 − 2.749115044705401165409263038133$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$F\left({z}^{3}\right)$ | 4.3807239279747234048593708927 − 1.57393791944848641246978433502$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$T{S}_{2}\left({z}^{3}\right)$ | −83.333333333333333333333333333 + 0 $\phantom{\rule{0.166667em}{0ex}}i$ | ||

$\pi /3$ | 2 | MB Int (M1 = 1) | 80.791062865366238781576251609 + 0 $\phantom{\rule{0.166667em}{0ex}}i$ |

Log. Term (M1 = 1) | 5.0698798575073995786757215377 − 1.56765473414130682599285904825$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Total via M1 | 6.9083333175150284317780107065 − 3.141592653589793238462643383279$\phantom{\rule{0.166667em}{0ex}}i$ | ||

LogGamma[zcube] | 6.9083333175150284317780107065 − 3.141592653589793238462643383279$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$F\left({z}^{3}\right)$ | 4.3671839976260822611773860371 + 0.45442183940812929747019906926$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$T{S}_{5}\left({z}^{3}\right)$ | −5.95238271839508333182790$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{17}$− 7.737535164228715974701668$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{11}\phantom{\rule{0.166667em}{0ex}}i$ | ||

$4\pi /7$ | 5 | MB Int (M1 = 1) | 5.95238271839508330650668$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{17}$ + 7.737535164239864652939712$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{11}\phantom{\rule{0.166667em}{0ex}}i$ |

${S}_{MB}(1,{z}^{3})$ | 5.0674216504983723676332001993 + 2.46643387314754360954113189663$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Total via M1 | 6.9024828161628933968353749191 + 4.03572353636012752353352062801$\phantom{\rule{0.166667em}{0ex}}i$ | ||

MB Int (M2 = 2) | 5.95238271839508335723002$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{17}$ + 7.737535164233152240154912$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{11}\phantom{\rule{0.166667em}{0ex}}i$ | ||

${S}_{MB}^{U}(2,{z}^{3})$ | 5.0710816111765935339415417904 − 1.175195955088607468412668235518$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Total via M2 | 6.9024828161628933880358773203 + 4.03572353636012787577347290435$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$F\left({z}^{3}\right)$ | 4.3749562509709981184827498273 − 1.05509306570630337542065838646$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$T{S}_{6}\left({z}^{3}\right)$ | 8.41751139369492541714725$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{23}$ + 5.154919990982005385545807$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{17}\phantom{\rule{0.166667em}{0ex}}i$ | ||

$8\pi /9$ | 6 | MB Int (M1 = 2) | −8.41751139369492541714722$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{23}$− 5.154919990982005395943833$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{17}\phantom{\rule{0.166667em}{0ex}}i$ |

${S}_{MB}^{U}(2,{z}^{3})$ | 0.0054413980927026535517822347 − 3.13845106093620344522418073989$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Total via M1 | 6.9134848732085864689307216827 − 5.23334675905750858730776717724$\phantom{\rule{0.166667em}{0ex}}i$ | ||

MB Int (M2 = 3) | −8.41751139369492541714727$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{23}$− 5.154919990982005390723539$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{17}\phantom{\rule{0.166667em}{0ex}}i$ | ||

${S}_{MB}^{U}(3,{z}^{3})$ | 5.0780394872445415442384564511 − 3.660480464761928202988399777482$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Total via M2 | 6.9134848732729541247531300647 − 5.233346759035781054140026255124$\phantom{\rule{0.166667em}{0ex}}i$ |

**Table 7.**$ln\mathsf{\Gamma}\left({z}^{3}\right)$ via the MB-regularized forms in the vicinity of the lines of discontinuity given by $\theta \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}-\pi /6$, $\theta \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\pi /2$ and $\theta \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}5\pi /6$ with $\left|z\right|\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1/10$ and $N\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}5$.

$\mathit{\theta}$ | Quantity | Value |
---|---|---|

$19\pi /40$ | Total via M1(=1) | 6.9017797138225092740511474835 − 1.3331484570580039616320161702$\phantom{\rule{0.166667em}{0ex}}i$ |

Total via M2(=2) | 6.9017797138225092740511474835 − 1.3331484570580039616320161702$\phantom{\rule{0.166667em}{0ex}}i$ | |

$\pi /2$ | Total via M1(=1) | 6.9014712712081946221027015741 − 1.5702191115306805133718396291$\phantom{\rule{0.166667em}{0ex}}i$ |

Total via M2(=2) | 6.9014712712081946221027015741 + 4.7129661956489059635534471374$\phantom{\rule{0.166667em}{0ex}}i$ | |

$21\pi /40$ | Total via M1(=1) | 6.9015102177011253639269574406 + 4.4758636443386950563445853873$\phantom{\rule{0.166667em}{0ex}}i$ |

Total via M2(=2) | 6.9015102177011253639269574406 + 4.4758636443386950563445853873$\phantom{\rule{0.166667em}{0ex}}i$ | |

$62\pi /75$ | Total via M1(=2) | 6.9139890062805982640446908483 + 1.6326576822112902043838680277$\phantom{\rule{0.166667em}{0ex}}i$ |

Total via M2(=3) | 6.9139890062805982640446908483 + 1.6326576822112902043838680277$\phantom{\rule{0.166667em}{0ex}}i$ | |

$5\pi /6$ | Total via M1(=2) | 6.9140376418225537950565521476 + 1.5702191115306805133718396291$\phantom{\rule{0.166667em}{0ex}}i$ |

Total via M2(=3) | 6.9140376418225537950565521476 + 1.5702191115306805133718396291$\phantom{\rule{0.166667em}{0ex}}i$ | |

$63\pi /75$ | Total via M1(=2) | 6.9140614934733956410110131643 − 4.7754024883371855095758498194$\phantom{\rule{0.166667em}{0ex}}i$ |

Total via M2(=3) | 6.9140614934733956410110131643 − 4.7754024883371855095758498194$\phantom{\rule{0.166667em}{0ex}}i$ | |

$-12\pi /75$ | Total via M1(=0) | 6.9077182194043652368591902822 + 1.5085404469087662793902283694$\phantom{\rule{0.166667em}{0ex}}i$ |

Total via M2(=−1) | 6.9077182194043652368591902822 + 1.5085404469087662793902283694$\phantom{\rule{0.166667em}{0ex}}i$ | |

$ln\mathsf{\Gamma}\left({z}^{3}\right)$ | 6.9077182194043652368591902822 + 1.5085404469087662793902283694$\phantom{\rule{0.166667em}{0ex}}i$ | |

$-\pi /6$ | Total via M1(=0) | 6.9077544565153742085796268609 + 1.5713735420591127250908037541$\phantom{\rule{0.166667em}{0ex}}i$ |

Total via M2(=−1) | 6.9077544565153742085796268609 + 1.5713735420591127250908037541$\phantom{\rule{0.166667em}{0ex}}i$ | |

$ln\mathsf{\Gamma}\left({z}^{3}\right)$ | 6.9077544565153742085796268609 + 1.5713735420591127250908037541$\phantom{\rule{0.166667em}{0ex}}i$ | |

$-13\pi /75$ | Total via M1(=0) | 6.9077907065971626138255125982 + 1.6342043592171290258017534223$\phantom{\rule{0.166667em}{0ex}}i$ |

Total via M2(=−1) | 6.9077907065971626138255125982 + 1.6342043592171290258017534223$\phantom{\rule{0.166667em}{0ex}}i$ | |

$ln\mathsf{\Gamma}\left({z}^{3}\right)$ | 6.9077907065971626138255125982 + 1.6342043592171290258017534223$\phantom{\rule{0.166667em}{0ex}}i$ |

**Table 8.**$ln\mathsf{\Gamma}\left({z}^{3}\right)$ for $\left|z\right|=5/2$ and various values of $\theta $ and N.

$\mathit{\theta}$ | N | Quantity | Value |
---|---|---|---|

$F\left({z}^{3}\right)$ | −14.88486001926988218316890967 − 30.6490797188237317042533096605$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$T{S}_{4}\left({z}^{3}\right)$ | 0.001187233093636153159546970 + 0.00520018521369322415111941337$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$-\pi /7$ | 4 | MB Int (M1 = 0) | 2.63163543021301667181503$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ − 6.6918000701164118190594$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}\phantom{\rule{0.166667em}{0ex}}i$ |

Log. Term (M1) | 0 | ||

Total via M1 | −14.88367278617361439457914968 − 30.6438795645051371993532040127$\phantom{\rule{0.166667em}{0ex}}i$ | ||

MB Int (M2 = −1) | 2.63163543021301667181503$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ − 6.6918000701164118190594$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}\phantom{\rule{0.166667em}{0ex}}i$ | ||

Log. Term (M2) | −2.67692877187577721034710$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-42}$− 3.9146393659521104430365$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-43}\phantom{\rule{0.166667em}{0ex}}i$ | ||

Total via M2 | −14.88367278617361439457914968 − 30.6438795645051371993532040127$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Borel Rem | 2.63163543021301667181503$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ − 6.6918000701164118190594$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}\phantom{\rule{0.166667em}{0ex}}i$ | ||

Borel Log Term | 0 | ||

Borel Total | −14.88367278617361439457914968 − 30.6438795645051371993532040127$\phantom{\rule{0.166667em}{0ex}}i$ | ||

LogGamma[zcube] | −14.88367278617361439457914968 − 30.6438795645051371993532040127$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$F\left({z}^{3}\right)$ | 26.8706304919944588244828769703 + 0 $\phantom{\rule{0.166667em}{0ex}}i$ | ||

$T{S}_{2}\left({z}^{3}\right)$ | 0.0053333333333333333333333333 + 0 $\phantom{\rule{0.166667em}{0ex}}i$ | ||

$2\pi /3$ | 2 | MB Int (M1 = 2) | −7.27328204587763887038193$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ + 0 $\phantom{\rule{0.166667em}{0ex}}i$ |

Log. Term (M1 = 2) | − 0.785398163397448309615660845819$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Total via M1 | 26.875963097999587570052547525 − 0.785398163397448309615660845819$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Borel Rem | −7.27328204587763662777752$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ + 0 $\phantom{\rule{0.166667em}{0ex}}i$ | ||

Borel Log Term | − 0.785398163397448309615660845819$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Borel Total | 26.875963097999587570052323265 − 0.785398163397448309615660845819$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$F\left({z}^{3}\right)$ | −45.81050523277279074549656662 − 7.88812400847329603213924107797$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$T{S}_{4}\left({z}^{3}\right)$ | −0.003771750463193365200048475 − 0.00377072066430421316496938343$\phantom{\rule{0.166667em}{0ex}}i$ | ||

$11\pi /12$ | 4 | MB Int (M1 = 2) | 1.8403031527099105137119$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ − 1.86174503255187256069532$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}\phantom{\rule{0.166667em}{0ex}}i$ |

Log Term (M1 = 2) | 69.42004590872447260962314046 − 2.83658512384077187501765734573$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Total via M1 | 23.60576892549032880207923528 − 10.7284798529802338653544196797$\phantom{\rule{0.166667em}{0ex}}i$ | ||

MB Int (M2 = 3) | 1.8403031527099105130346$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ − 1.86174503255187256048211$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}\phantom{\rule{0.166667em}{0ex}}i$ | ||

Log Term (M2 = 3) | 69.42004590872447260962314046 − 2.83658512384077187501765734573$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Total via M2 | 23.60576892549032880207923528 − 10.7284798529802338653544196797$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Borel Rem | 1.8403031527099105130346$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ − 1.86174503255187256048211$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}\phantom{\rule{0.166667em}{0ex}}i$ | ||

Borel Log Term | 69.42004590872447260962314046 − 2.83658512384077187501765734573$\phantom{\rule{0.166667em}{0ex}}i$ | ||

Borel Total | 23.60576892549032880207923528 − 10.7284798529802338653544196797$\phantom{\rule{0.166667em}{0ex}}i$ |

**Table 9.**$ln\mathsf{\Gamma}\left({z}^{3}\right)$ for $\left|z\right|=9/10$ at the Stokes lines within the principal branch.

$\mathit{\theta}$ | Method | Value |
---|---|---|

(40) | −0.0629795852996006019126614 − 1.86781980997058048039434088$\phantom{\rule{0.166667em}{0ex}}i$ | |

$\pi /6$ | (44) | −0.0629795852996006019126614 − 1.86781980997058048039434088$\phantom{\rule{0.166667em}{0ex}}i$ |

Top, (58) | −0.0629795852996006019126614 − 1.86781980997058048039434088$\phantom{\rule{0.166667em}{0ex}}i$ | |

LogGamma[z] | −0.0629795852996006019126614 − 1.86781980997058048039434088$\phantom{\rule{0.166667em}{0ex}}i$ | |

(44), c.c. | −4.6434216742335191435911954 − 1.86781980997058048039434088$\phantom{\rule{0.166667em}{0ex}}i$ | |

$-\pi /2$ | Bottom, (47), c.c. | −4.6434216742335191435911954 − 1.86781980997058048039434088$\phantom{\rule{0.166667em}{0ex}}i$ |

Middle, (58) | −4.6434216742335191435911954 − 1.86781980997058048039434088$\phantom{\rule{0.166667em}{0ex}}i$ | |

Top, (47) | 4.5174625036343179397658872 − 1.86781980997058048039434088$\phantom{\rule{0.166667em}{0ex}}i$ | |

$5\pi /6$ | Bottom, (47) | 4.5174625036343179397658872 − 1.86781980997058048039434088$\phantom{\rule{0.166667em}{0ex}}i$ |

Bottom, (58) | 4.5174625036343179397658872 − 1.86781980997058048039434088$\phantom{\rule{0.166667em}{0ex}}i$ |

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**MDPI and ACS Style**

Kowalenko, V.
Exact Values of the Gamma Function from Stirling’s Formula. *Mathematics* **2020**, *8*, 1058.
https://doi.org/10.3390/math8071058

**AMA Style**

Kowalenko V.
Exact Values of the Gamma Function from Stirling’s Formula. *Mathematics*. 2020; 8(7):1058.
https://doi.org/10.3390/math8071058

**Chicago/Turabian Style**

Kowalenko, Victor.
2020. "Exact Values of the Gamma Function from Stirling’s Formula" *Mathematics* 8, no. 7: 1058.
https://doi.org/10.3390/math8071058