Particular values of the gamma function

Integers and half-integers

For positive integer arguments, the gamma function coincides with the factorial. That is,

:\Gamma(n) = (n-1)!,

and hence

:\begin{align} \Gamma(1) &= 1, \ \Gamma(2) &= 1, \ \Gamma(3) &= 2, \ \Gamma(4) &= 6, \ \Gamma(5) &= 24, \end{align}

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

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

:\Gamma \left (\tfrac{k}{2} \right) = \sqrt \pi \frac{(k-2)!!}{2^\frac{k-1}{2}},,

or equivalently, for non-negative integer values of n:

:\begin{align} \Gamma\left(\tfrac12+n\right) &= \frac{(2n-1)!!}{2^n}, \sqrt{\pi} = \frac{(2n)!}{4^n n!} \sqrt{\pi} \ \Gamma\left(\tfrac12-n\right) &= \frac{(-2)^n}{(2n-1)!!}, \sqrt{\pi} = \frac{(-4)^n n!}{(2n)!} \sqrt{\pi} \end{align}

where n!! denotes the double factorial. In particular,

| |- |\Gamma\left(\tfrac32\right), |= \tfrac12 \sqrt{\pi}, |\approx 0.886,226,925,452,758,0137,, | |- |\Gamma\left(\tfrac52\right), |= \tfrac34 \sqrt{\pi}, |\approx 1.329,340,388,179,137,0205,, | |- |\Gamma\left(\tfrac72\right), |= \tfrac{15}8 \sqrt{\pi}, |\approx 3.323,350,970,447,842,5512,, | |}

and by means of the reflection formula,

| |- |\Gamma\left(-\tfrac32\right), |= \tfrac43 \sqrt{\pi}, |\approx 2.363,271,801,207,354,7031,, | |- |\Gamma\left(-\tfrac52\right), |= -\tfrac 8{15} \sqrt{\pi}, |\approx -0.945,308,720,482,941,8812,, | |}

General rational argument

In analogy with the half-integer formula,

:\begin{align} \Gamma \left(n+\tfrac13 \right) &= \Gamma \left(\tfrac13 \right) \frac{(3n-2)!!!}{3^n} \ \Gamma \left(n+\tfrac14 \right) &= \Gamma \left(\tfrac14 \right ) \frac{(4n-3)!!!!}{4^n} \ \Gamma \left(n+\tfrac{1}{q} \right ) &= \Gamma \left(\tfrac{1}{q} \right ) \frac{\big(qn-(q-1)\big)!^{(q)}}{q^n} \ \Gamma \left(n+\tfrac{p}{q} \right) &= \Gamma \left(\tfrac{p}{q}\right) \frac{1}{q^n} \prod _{k=1}^n (k q+p-q) \end{align}

where n!(q) denotes the qth multifactorial of n. Numerically,

:\Gamma\left(\tfrac13\right) \approx 2.678,938,534,707,747,6337 :\Gamma\left(\tfrac14\right) \approx 3.625,609,908,221,908,3119 :\Gamma\left(\tfrac15\right) \approx 4.590,843,711,998,803,0532 :\Gamma\left(\tfrac16\right) \approx 5.566,316,001,780,235,2043 :\Gamma\left(\tfrac17\right) \approx 6.548,062,940,247,824,4377 :\Gamma\left(\tfrac18\right) \approx 7.533,941,598,797,611,9047 .

Additionally,

:\lim_{n\to\infty}\left(n-\Gamma\left(\tfrac1n\right)\right) =\gamma

where \gamma is the Euler–Mascheroni constant.

It is unknown whether these constants are transcendental in general, but Γ() and Γ() were shown to be transcendental by G. V. Chudnovsky. Γ() / has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that Γ(), π, and eπ are algebraically independent.

For n\geq 2 at least one of the two numbers \Gamma\left(\tfrac1n\right) and \Gamma\left(\tfrac2n\right) is transcendental.

The number \Gamma\left(\tfrac14\right) is related to the lemniscate constant \varpi by

:\Gamma\left(\tfrac14\right) = \sqrt{2\varpi\sqrt{2\pi}}

Borwein and Zucker have found that Γ() can be expressed algebraically in terms of π, K(k(1)), K(k(2)), K(k(3)) and K(k(6)) where 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 quadratically convergent arithmetic–geometric mean iterations. For example: :\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{1}{\sqrt{2}} \right)\sqrt{\pi}} \ \Gamma \left(\tfrac13 \right) &= \frac{2^{7/9} \sqrt[3]{\pi K\left(\frac{\sqrt3-1}{2\sqrt2}\right)}}{\sqrt[12]{3}} \ \Gamma \left(\tfrac{1}{8}\right) \Gamma \left(\tfrac{3}{8}\right) &= 8 \sqrt[4]{2} \sqrt{\left(\sqrt{2}-1\right) \pi } K\left(3-2 \sqrt{2}\right) \ \frac{\Gamma \left(\frac{1}{8}\right)}{\Gamma \left(\frac{3}{8}\right)} &= \frac{2 \sqrt{\left(1+\sqrt{2}\right) K\left(\frac{1}{2}\right)}}{\sqrt[4]{\pi }} \end{align}

No similar relations are known for Γ() or other denominators.

In particular, where AGM() is the arithmetic–geometric mean, we have :\Gamma\left(\tfrac13\right) = \frac{2^\frac{1}{9}(2\pi)^\frac23}{3^\frac{1}{12}\cdot \operatorname{AGM}\left(2,\sqrt{2+\sqrt{3}}\right)^\frac13} :\Gamma\left(\tfrac14\right) = \sqrt \frac{(2 \pi)^\frac32}{\operatorname{AGM}\left(\sqrt 2, 1\right)} :\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}.

Other formulas include the infinite products

:\Gamma\left(\tfrac14\right) = (2 \pi)^\frac34 \prod_{k=1}^\infty \tanh \left( \frac{\pi k}{2} \right)

and

:\Gamma\left(\tfrac14\right) = A^3 e^{-\frac{G}{\pi}} 2^\frac16\sqrt{\pi} \prod_{k=1}^\infty \left(1-\frac{1}{2k}\right)^{k(-1)^k}

where A is the Glaisher–Kinkelin constant and G is Catalan's constant.

The following two representations for Γ() were given by I. Mező{{Citation | doi-access = free

:\sqrt{\frac{\pi\sqrt{e^\pi}}{2}}\frac{1}{\Gamma\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),

and

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

where θ1 and θ4 are two of the Jacobi theta functions.

There also exist a number of Malmsten integrals for certain values of the gamma function:

:\int_1^\infty \frac{\ln \ln t}{1+t^2} = \frac\pi4\left(2\ln2 + 3\ln\pi-4\Gamma\left(\tfrac14\right)\right) :\int_1^\infty \frac{\ln \ln t}{1+t+t^2} = \frac\pi{6\sqrt3}\left(8\ln2\pi -3\ln3 -12\Gamma\left(\tfrac13\right)\right)

Products

Some product identities include:

: \prod_{r=1}^2 \Gamma\left(\tfrac{r}{3}\right) = \frac{2\pi}{\sqrt 3} \approx 3.627,598,728,468,435,7012 : \prod_{r=1}^3 \Gamma\left(\tfrac{r}{4}\right) = \sqrt{2\pi^3} \approx 7.874,804,972,861,209,8721 : \prod_{r=1}^4 \Gamma\left(\tfrac{r}{5}\right) = \frac{4\pi^2}{\sqrt 5} \approx 17.655,285,081,493,524,2483 : \prod_{r=1}^5 \Gamma\left(\tfrac{r}{6}\right) = 4\sqrt{\frac{\pi^5}3} \approx 40.399,319,122,003,790,0785 : \prod_{r=1}^6 \Gamma\left(\tfrac{r}{7}\right) = \frac{8\pi^3}{\sqrt 7} \approx 93.754,168,203,582,503,7970 : \prod_{r=1}^7 \Gamma\left(\tfrac{r}{8}\right) = 4\sqrt{\pi^7} \approx 219.828,778,016,957,263,6207

In general: : \prod_{r=1}^n \Gamma\left(\tfrac{r}{n+1}\right) = \sqrt{\frac{(2\pi)^n}{n+1}}

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

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

Other rational relations include

:\frac{\Gamma\left(\tfrac15\right)\Gamma\left(\tfrac{4}{15}\right)}{\Gamma\left(\tfrac13\right)\Gamma\left(\tfrac{2}{15}\right)} = \frac{\sqrt{2},\sqrt[20]{3}}{\sqrt[6]{5},\sqrt[4]{5-\frac{7}{\sqrt 5}+\sqrt{6-\frac{6}{\sqrt 5}}}}

:\frac{\Gamma\left(\tfrac{1}{20}\right)\Gamma\left(\tfrac{9}{20}\right)}{\Gamma\left(\tfrac{3}{20}\right)\Gamma\left(\tfrac{7}{20}\right)} = \frac{\sqrt[4]{5}\left(1+\sqrt{5}\right)}{2}

:\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}}

and many more relations for Γ() where the denominator d divides 24 or 60.

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.

A more sophisticated example: : \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}}

Imaginary and complex arguments

The gamma function at the imaginary unit gives , : :\Gamma(i) = (-1+i)! \approx -0.1549 - 0.4980i. It may also be given in terms of the Barnes G-function:

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

\Gamma(i) appears in the below integral evaluation: :\int_0^{\pi/2}{\cot(x)},dx=1-\frac{\pi}{2}+\frac{i}{2}\log\left(\frac{\pi}{\sinh(\pi)\Gamma(i)^2}\right). Here {\cdot} denotes the fractional part.

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

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

The above integral therefore relates to the phase of \Gamma(i).

The gamma function with other complex arguments returns :\Gamma(1 + i) = i\Gamma(i) \approx 0.498 - 0.155i :\Gamma(1 - i) = -i\Gamma(-i) \approx 0.498 + 0.155i :\Gamma(\tfrac12 + \tfrac12 i) \approx 0.818,163,9995 - 0.763,313,8287, i :\Gamma(\tfrac12 - \tfrac12 i) \approx 0.818,163,9995 + 0.763,313,8287, i :\Gamma(5 + 3i) \approx 0.016,041,8827 - 9.433,293,2898, i :\Gamma(5 - 3i) \approx 0.016,041,8827 + 9.433,293,2898, i.

Other constants

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

:x_{\min} = 1.461,632,144,968,362,341,262,659,5423\ldots,

with the value

:\Gamma\left(x_{\min}\right) = 0.885,603,194,410,888,700,278,815,9005\ldots, .

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

On the negative real axis, the first local maxima and minima (zeros of the digamma function) are:

The only values of x 0 for which are and x ≈ ... .

References

References

1. Waldschmidt, Michel. (2006). "Transcendence of periods: the state of the art". Pure and Applied Mathematics Quarterly.
2. "Archived copy".
3. Blagouchine, Iaroslav V.. (2014-10-01). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results". The Ramanujan Journal.
4. {{MathWorld. GammaFunction. Gamma Function
5. [https://arxiv.org/abs/math/0403510 Raimundas Vidūnas, Expressions for Values of the Gamma Function]
6. [https://math.stackexchange.com/q/2804457 math.stackexchange.com]
7. [https://sites.google.com/site/istvanmezo81/monthly-problems The webpage of István Mező]