Particular values of the gamma function: Difference between revisions

Revision as of 12:00, 20 August 2024

Latest revision as of 06:26, 15 July 2025

(19 intermediate revisions by 7 users not shown)

Line 1:

{{Short description|Mathematical constants}}

The [[gamma function]] is an important [[special function]] in [[mathematics]]. Its particular values can be expressed in closed form for [[integer]] and, [[half-integer]], and some other rational arguments, but no simple expressions are known for the values at [[rational number|rational]] points in general. Other fractional arguments can be approximated through efficient infinite products, infinite series, and recurrence relations.

==Integers and half-integers==

Line 19:

and so on. For non-positive integers, the gamma function is not defined.

For positive half-integers <math> \frac{k}{2} </math> where <math> k\in 2\mathbb{N}^*+1 </math> is an odd integer greater or equal <math>3</math>, the function values are given exactly by

:<math>\Gamma \left (\tfrac{nk}{2} \right) = \sqrt \pi \frac{(nk-2)!!}{2^\frac{nk-1}{2}}\,,</math>

or equivalently, for non-negative integer values of&nbsp;{{mvar|n}}:

Line 85:

\end{align}</math>

where {{math|''n''!^(''q'')}} denotes the {{mvar|q}}th [[Double factorialDouble_factorial#Generalizations|multifactorial]] of {{mvar|n}}. Numerically,

:<math>\Gamma\left(\tfrac13\right) \approx 2.678\,938\,534\,707\,747\,6337</math> {{OEIS2C|A073005}}

Line 96:

As <math>n</math> tends to infinity,

:<math>\Gamma\left(\tfrac1n\right) \sim n-\gamma </math>

where <math>\gamma</math> is the [[Euler–Mascheroni constant]] and <math>\sim</math> denotes [[Asymptotic analysis|asymptotic equivalence]].

It is unknown whether these constants are [[transcendental number|transcendental]] in general, but <{{math>\Gamma\left|Γ(\tfrac13\right{{sfrac|1|3}})</math>}} and <{{math>\Gamma\left|Γ(\tfrac14\right{{sfrac|1|4}})</math>}} were shown to be transcendental by [[Chudnovsky brothers|G. V. Chudnovsky]]. <{{math>\pi^|Γ({-\frac14{sfrac|1|4}}\Gamma\left(\tfrac14\right) <big><big>/math</big></big> {{radic|π|4}}}} has also long been known to be transcendental, and [[Yuri Valentinovich Nesterenko|Yuri Nesterenko]] proved in 1996 that <{{math>\Gamma\left|Γ(\tfrac14\right{{sfrac|1|4}})}}, \pi</{{math>|π}}, and <{{math>|''e^\pi''<sup>π</mathsup>}} are [[algebraically independent]].

It is known that forFor <math>n\geq 32</math>  at least one of the two numbers  <math>\Gamma\left(\tfrac1n\right)</math> and <math>\Gamma\left(\tfrac2n\right)</math>  is transcendental.<ref>{{Cite journal |last=Waldschmidt |first=Michel |date=2006 |title=Transcendence of periods: the state of the art |url=https://hal.science/hal-00411301 |journal=Pure and Applied Mathematics Quarterly |volume=2 |issue=2 |pages=435–463|doi=10.4310/PAMQ.2006.v2.n2.a3 }}</ref>

The number <math>\Gamma\left(\tfrac14\right)</math> is related to the [[lemniscate constant]] {{mvar|<math>\varpi</math>}} by

:<math>\Gamma\left(\tfrac14\right) = \sqrt{2\varpi\sqrt{2\pi}},</math>

Borwein and Zucker have found that {{math|Γ({{sfrac|''n''|24}})}} can be expressed algebraically in terms of {{mvar|π}}, {{math|''K''(''k''(1))}}, {{math|''K''(''k''(2))}}, {{math|''K''(''k''(3))}}, and {{math|''K''(''k''(6))}} where {{math|''K''(''k''(''N''))}} is a [[complete elliptic integral of the first kind]]. This permits efficiently approximating the gamma function of rational arguments to high precision using [[quadratic convergence|quadratically convergent]] [[arithmetic–geometric mean]] iterations. For example:

Borwein and Zucker<ref>{{Cite journal

|first1=J. M.

|last1=Borwein

|first2=I. J.

|last2=Zucker

|title=Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind

|journal=IMA Journal of Numerical Analysis

|volume=12

|issue=4

|pages=519&ndash;526

|year=1992

|mr=1186733

|doi=10.1093/imanum/12.4.519

}}</ref> have found that <math>\Gamma\left(\tfrac{n}{24}\right)</math> can be expressed algebraically in terms of {{mvar|π}}, {{math|''K''(''k''(1))}}, {{math|''K''(''k''(2))}}, {{math|''K''(''k''(3))}}, and {{math|''K''(''k''(6))}} where {{math|''K''(''k''(''N''))}} is a [[complete elliptic integral of the first kind]]. This permits efficiently approximating the gamma function of rational arguments to high precision using [[quadratic convergence|quadratically convergent]] [[arithmetic–geometric mean]] iterations. For example:

:<math>\begin{align}

\Gamma \left(\tfrac16 \right) &= \frac{\sqrt{\frac{3}{\pi }} \Gamma\left(\frac{1}{3}\right)^2}{\sqrt[3]{2}} \\

\Gamma \left(\tfrac14 \right) &= 2\sqrt{K\left( \tfrac 12 \right)\sqrt{\pi}} \\

\Gamma \left(\tfrac13 \right) &= \frac{2^{7/9} \sqrt[3]{\pi K\left(\frac{1}{4}\left(2-\sqrt{3}\right)\right)}}{\sqrt[12]{3}} \\

\Gamma \left(\tfrac 18{1}{8}\right) \Gamma \left(\tfrac 38{3}{8}\right) &= 8 \sqrt[4]{2} \sqrt{\left(\sqrt{2}-1\right) \pi } K\left(3-2 \sqrt{2}\right) \\

\frac{\Gamma \left(\tfrac 18frac{1}{8}\right)}{\Gamma \left(\tfrac 38frac{3}{8}\right)} &= \frac{2 \sqrt{\left(1+\sqrt{2}\right) K\left(\frac{1}{2}\right)}}{\sqrt[4]{\pi }}

\end{align}</math>

No similar relations are known for {{math|Γ({{sfrac|1|5}})}} or other denominators.

In particular, where AGM() is the [[arithmetic–geometric mean]], we have<ref>{{cite web|url=https://math.stackexchange.com/q/1631760 |title=Archived copy |accessdate=2015-03-09 }}</ref>

:<math>\Gamma\left(\tfrac13\right) = \frac{2^\frac{7}{9}\cdot \pi^\frac23}{3^\frac{1}{12}\cdot \operatorname{AGM}\left(2,\sqrt{2+\sqrt{3}}\right)^\frac13}</math>

:<math>\Gamma\left(\tfrac14\right) = \sqrt \frac{(2 \pi)^\frac32}{\operatorname{AGM}\left(\sqrt 2, 1\right)}</math>

:<math>\Gamma\left(\tfrac16\right) = \frac{2^\frac{14}{9}\cdot 3^\frac13\cdot \pi^\frac56}{\operatorname{AGM}\left(1+\sqrt{3},\sqrt{8}\right)^\frac23}.</math>

Other formulas include the [[infinite product]]s

Line 140 ⟶ 134:

where {{mvar|A}} is the [[Glaisher–Kinkelin constant]] and {{mvar|G}} is [[Catalan's constant]].

The following two representations for <{{math>\Gamma\left|Γ(\tfrac34\right{{sfrac|3|4}})</math>}} were given by I. Mező<ref name="Mezo2">{{Citation

| last = Mező

| first = István

Line 153 ⟶ 147:

}}</ref>

:<math>\sqrt{\frac{\pi\sqrt{e^\pi}}{2}}\frac{1}{\Gamma^2\left(\frac34\right)^2}=i\sum_{k=-\infty}^\infty e^{\pi(k-2k^2)}\theta_1\left(\frac{i\pi}{2}(2k-1),e^{-\pi}\right),</math>

and

:<math>\sqrt{\frac{\pi}{2}}\frac{1}{\Gamma^2\left(\frac34\right)^2}=\sum_{k=-\infty}^\infty\frac{\theta_4(ik\pi,e^{-\pi})}{e^{2\pi k^2}},</math>

where {{math|''θ''₁}} and {{math|''θ''₄}} are two of the [[Theta function|Jacobi theta functions]].

There also exist a number of [[Carl Johan Malmsten|Malmsten integrals]] for certain values of the gamma function:<ref name=":1">{{Cite journal |last=Blagouchine |first=Iaroslav V. |date=2014-10-01 |title=Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results |url=https://link.springer.com/article/10.1007/s11139-013-9528-5 |journal=The Ramanujan Journal |language=en |volume=35 |issue=1 |pages=21–110 |doi=10.1007/s11139-013-9528-5 |issn=1572-9303|url-access=subscription }}</ref>

Certain values of the gamma function can also be written in terms of the [[hypergeometric function]]. For instance, <math>\Gamma\left(\frac{1}{4}\right)^{4}=\frac{32\pi^{3}}{\sqrt{33}} {}_{3}F_{2}\left(\frac{1}{2},\ \frac{1}{6},\ \frac{5}{6};\ 1,\ 1;\ \frac{8}{1331}\right)</math>

:<math>\int_1^\infty \frac{\ln \ln t}{1+t^2} = \frac\pi4\left(2\ln2 + 3\ln\pi-4\Gamma\left(\tfrac14\right)\right)</math>

and

:<math>\int_1^\infty \frac{\ln \ln t}{1+t+t^2} = \frac\pi{6\sqrt3}\left(8\ln2 -3\ln3 + 8\ln\pi -12\Gamma\left(\tfrac13\right)\right)</math>

<math>\Gamma\left(\frac{1}{3}\right)^{6}=\frac{12\pi^{4}}{\sqrt{10}} {}_{3}F_{2}\left(\frac{1}{2},\ \frac{1}{6},\ \frac{5}{6};\ 1,\ 1;\ -\frac{9}{64000}\right)</math>

however it is an open question whether this is possible for all rational inputs to the gamma function.<ref>Johansson, F. (2023). Arbitrary-precision computation of the gamma function. ''Maple Transactions'', ''3''(1). {{doi|10.5206/mt.v3i1.14591}}</ref>

== Products ==

Line 181 ⟶ 172:

In general:

:<math> \prod_{r=1}^n \Gamma\left(\tfrac{r}{n+1}\right) = \sqrt{\frac{(2\pi)^n}{n+1}}</math>

From those products can be deduced other values, for example, from the former equations for <math> \prod_{r=1}^3 \Gamma\left(\tfrac{r}{4}\right) </math>, <math>\Gamma\left(\tfrac{1}{4}\right) </math> and <math>\Gamma\left(\tfrac{2}{4}\right) </math>, can be deduced:

<math>\Gamma\left(\tfrac{3}{4}\right) =\left(\tfrac{\pi} {2}\right) ^{\tfrac{1}{4}} {\operatorname{AGM}\left(\sqrt 2, 1\right)}^{\tfrac{1}{2}}</math>

Other rational relations include

Line 190 ⟶ 185:

:<math>\frac{\Gamma\left(\frac{1}{5}\right)^2}{\Gamma\left(\frac{1}{10}\right)\Gamma\left(\frac{3}{10}\right)} = \frac{\sqrt{1+\sqrt{5}}}{2^{\tfrac{7}{10}}\sqrt[4]{5}}</math>

and many more relations for <{{math>\Gamma\left|Γ(\tfrac{{sfrac|''n}{''|''d''}}\right)</math>}} where the denominator d divides 24 or 60.<ref>{{cite[https://arxiv.org/abs/math/0403510 journalRaimundas Vidūnas, Expressions for Values of the Gamma Function]</ref>

| last = Vidūnas | first = Raimundas

Gamma quotients with algebraic values must be "poised" in the sense that the sum of arguments is the same (modulo 1) for the denominator and the numerator.

| arxiv = math/0403510

| doi = 10.2206/kyushujm.59.267

A more sophisticated example:

| issue = 2

:<math> \frac{ \Gamma\left(\frac{11}{42}\right)\Gamma\left(\frac27\right)}{\Gamma\left(\frac1{21}\right)\Gamma\left(\frac1{2}\right)} = \frac{8 \sin\left(\frac\pi7\right) \sqrt{\sin\left(\frac\pi{21}\right) \sin\left(\frac{4\pi}{21}\right) \sin\left(\frac{5\pi}{21}\right)}}{2^{\frac1{42}}3^{\frac9{28}}7^{\frac13}} </math><ref>[https://math.stackexchange.com/q/2804457 math.stackexchange.com]</ref>

| journal = Kyushu Journal of Mathematics

| mr = 2188592

| pages = 267–283

| title = Expressions for values of the gamma function

| volume = 59

| year = 2005}}</ref>

== Imaginary and complex arguments==

Line 209 ⟶ 199:

:<math>\Gamma(i) = \frac{G(1+i)}{G(i)} = e^{-\log G(i)+ \log G(1+i)}.</math>

Curiously enough, <math>\Gamma(i)</math> appears in the below integral evaluation:<ref>[https://sites.google.com/site/istvanmezo81/monthly-problems The webpage of István Mező]</ref>

Because of the [[Reflection formula|Euler Reflection Formula]], and the fact that <math>\Gamma(\overline{z})=\overline{\Gamma(z)}</math>, we have an expression for the [[modulus squared]] of the gamma function evaluated on the imaginary axis:

:<math>\int_0^{\pi/2}\{\cot(x)\}\,dx=1-\frac{\pi}{2}+\frac{i}{2}\log\left(\frac{\pi}{\sinh(\pi)\Gamma(i)^2}\right).</math>

Here <math>\{\cdot\}</math> denotes the [[fractional part]].

Because of the [[Reflection formula|Euler Reflection Formula]], and the fact that <math>\Gamma(\bar{z})=\bar{\Gamma}(z)</math>, we have an expression for the [[modulus squared]] of the Gamma function evaluated on the imaginary axis:

:<math>\left|\Gamma(i\kappa)\right|^2=\frac{\pi}{\kappa\sinh(\pi\kappa)}</math>

Line 226 ⟶ 220:

The gamma function has a [[local minimum]] on the positive real axis

:<math>x_{\min} = 1.461\,632\,144\,968\,362\,341\,262\,659\,5423\ldots\,</math> {{OEIS2C|A030169}}

with the value

:<math>\Gamma\left(x_{\min}\right) = 0.885\,603\,194\,410\,888\,700\,278\,815\,9005\ldots\,</math> {{OEIS2C|A030171}}.

Integrating the [[reciprocal gamma function]] along the positive real axis also gives the [[Fransén–Robinson constant]].

Line 260 ⟶ 254:

| {{val|−9.7026725400018637360844267649}} || {{0|−}}{{val|0.0000021574161045228505405031}} || {{OEIS2C|A256687}}

|}

The only values of {{math|''x'' > 0}} for which {{math|1=Γ(''x'') = ''x''}} are {{math|1=''x'' = 1}} and {{math|''x'' ≈ {{val|3.5623822853908976914156443427}}}}... {{OEIS2C|A218802}}.

==See also==

Line 265 ⟶ 261:

==References==

<references />

{{reflist}}

==Further reading==

* {{Cite journal

|first1=F.

|last1=Gramain

|title=Sur le théorème de Fukagawa-Gel'fond

|journal=Invent. Math.

|volume=63

|number=3

|doi=10.1007/BF01389066

|year=1981

|pages=495&ndash;506

|bibcode=1981InMat..63..495G

|s2cid=123079859

}}

* {{Cite journal

|first1=J. M.

|last1=Borwein

|first2=I. J.

|last2=Zucker

|title=Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind

|journal=IMA Journal of Numerical Analysis

|volume=12

|issue=4

|pages=519&ndash;526

|year=1992

|mr=1186733

|doi=10.1093/imanum/12.4.519

}}

* X. Gourdon & P. Sebah. [http://numbers.computation.free.fr/Constants/Miscellaneous/gammaFunction.html Introduction to the Gamma Function]

* {{MathWorld|title=Gamma Function|urlname=GammaFunction}}

* {{cite journal

|first1=Raimundas |last1=Vidunas

|title=Expressions for values of the gamma function

|arxiv=math.CA/0403510

|doi=10.2206/kyushujm.59.267

|volume=59

|issue=2

|journal=Kyushu Journal of Mathematics

|pages=267–283|year=2005

|s2cid=119623635

}}

* {{cite journal

|first1=Raimundas | last1=Vidunas

|title=Expressions for values of the gamma function

|journal=Kyushu J. Math. |year=2005

|volume=59 | number=2 | pages=267–283 | mr=2188592 | doi=10.2206/kyushujm.59.267|arxiv=math/0403510| s2cid=119623635

}}

* {{Cite journal

|first1=V. S.

|last1=Adamchik

|url=https://www.researchgatecs.netcmu.edu/publication~adamchik/2108174articles/rama.pdf

|title=Multiple Gamma Function and Its Application to Computation of Series

|journal=The Ramanujan Journal