Particular values of the gamma function explained

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

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, the function values are given exactly by

\Gamma \left (\tfrac{n}{ 2 } \right) = \sqrt \pi ((n -2 )!!)/(2 ^{(n -1)/(2)}) ,

or equivalently, for non-negative integer values of :

\begin{align} \Gamma\left(\tfrac 12+ n\right) & = (( 2 n -1 )!!)/(2 ^{n}) \sqrt{\pi} = (( 2 n)!)/(4 ^{n} n!) \sqrt{\pi} \\ \Gamma\left(\tfrac 12- n\right) & = (( -2 ) ^{n})/(( 2 n -1 )!!) \sqrt{\pi} = (( -4 ) ^{n} n!)/(( 2 n)!) \sqrt{\pi} \end{align}

where denotes the double factorial. In particular,

\Gamma\left(\tfrac 12 \right)	= \sqrt{\pi}	≈ 1 . 7724538509055160273 ,
\Gamma\left(\tfrac 32 \right)	= \tfrac 12 \sqrt{\pi}	≈ 0 . 8862269254527580137 ,
\Gamma\left(\tfrac 52 \right)	= \tfrac 34 \sqrt{\pi}	≈ 1 . 3293403881791370205 ,
\Gamma\left(\tfrac 72 \right)	= \tfrac{ 15 } 8 \sqrt{\pi}	≈ 3 . 3233509704478425512 ,

and by means of the reflection formula,

\Gamma\left( - \tfrac 12 \right)	= -2 \sqrt{\pi}	≈ -3 . 5449077018110320546 ,
\Gamma\left( - \tfrac 32 \right)	= \tfrac 43 \sqrt{\pi}	≈ 2 . 3632718012073547031 ,
\Gamma\left( - \tfrac 52 \right)	= - \tfrac 8 { 15 } \sqrt{\pi}	≈ -0 . 9453087204829418812 ,

General rational argument

In analogy with the half-integer formula,

\begin{align} \Gamma \left(n + \tfrac 13 \right) & = \Gamma \left(\tfrac 13 \right) (( 3 n -2 )!!!)/(3 ^{n}) \\ \Gamma \left(n + \tfrac 14 \right) & = \Gamma \left(\tfrac 14 \right ) (( 4 n -3 )!!!!)/(4 ^{n}) \\ \Gamma \left(n + \tfrac{ 1 }{q} \right ) & = \Gamma \left(\tfrac{ 1 }{q} \right ) (( qn - (q -1 ) ) ! ^{(q)})/(q ^{n}) \\ \Gamma \left(n + \tfrac{p}{q} \right) & = \Gamma \left(\tfrac{p}{q}\right) (1)/(q ^{n}) \prod ^{n} ^{k =1} (k q + p - q) \end{align}

where denotes the th multifactorial of . Numerically,

\Gamma\left(\tfrac 13 \right) ≈ 2 . 6789385347077476337 \Gamma\left(\tfrac 14 \right) ≈ 3 . 6256099082219083119 \Gamma\left(\tfrac 15 \right) ≈ 4 . 5908437119988030532 \Gamma\left(\tfrac 16 \right) ≈ 5 . 5663160017802352043 \Gamma\left(\tfrac 17 \right) ≈ 6 . 5480629402478244377 \Gamma\left(\tfrac 18 \right) ≈ 7 . 5339415987976119047 .

As

n tends to infinity,\Gamma\left(\tfrac 1 n\right) \sim n - \gamma

where

\gamma is the Euler–Mascheroni constant and \sim denotes asymptotic equivalence.

It is unknown whether these constants are transcendental in general, but

\Gamma\left(\tfrac 13 \right) and \Gamma\left(\tfrac 14 \right) were shown to be transcendental by G. V. Chudnovsky. \pi ^{- (14)/()} \Gamma\left(\tfrac 14 \right) has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that \Gamma\left(\tfrac 14 \right), \pi and e ^{\pi} are algebraically independent.

It is known that for

n\geq 3 at least one of the two numbers \Gamma\left(\tfrac 1 n\right) and \Gamma\left(\tfrac 2 n\right) is transcendental.^[1]

The number

\Gamma\left(\tfrac 14 \right) is related to the lemniscate constant \varpi by\Gamma\left(\tfrac 14 \right) = \sqrt{ 2 \varpi\sqrt{ 2 \pi}},

Borwein and Zucker^[2] have found that

\Gamma\left(\tfrac{n}{ 24 }\right) can be expressed algebraically in terms of,,,, and where 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(\tfrac 16 \right) & = (\sqrt{ (3)/(\pi))/(\Gamma\left( (1)/(3)) \right) ^{2}{\sqrt[3]{2}}} \\ \Gamma \left(\tfrac 14 \right) & = 2 \sqrt{K\left( \tfrac 12 \right)\sqrt{\pi}} \\ \Gamma \left(\tfrac 13 \right) & = (2 ^{7 / 9} \sqrt[ 3 ]{\pi K\left( (1)/(4) \left( 2- \sqrt{ 3)/(\right)\right)}}{\sqrt[ 12 ]{ 3 }}) \\ \Gamma \left(\tfrac 18 \right) \Gamma \left(\tfrac 38 \right) & = 8 \sqrt[ 4 ]{ 2 } \sqrt{\left(\sqrt{ 2 } -1 \right) \pi } K\left( 3-2 \sqrt{ 2 }\right) \\ (\Gamma \left(\tfrac 18 \right))/(\Gamma \left(\tfrac 38 \right)) & = (2 \sqrt{\left( 1+ \sqrt{ 2)/(K\left( \right)) (1)/(2) \right)}}{\sqrt[ 4 ]{\pi }} \end{align}

Other formulas include the infinite products

\Gamma\left(\tfrac 14 \right) = ( 2 \pi) ^{inf ty} _{k =1} \tanh \left( (\pi k)/(2) \right)

and

\Gamma\left(\tfrac 14 \right) = A ^{3} e ^{- (G)/(\pi)} \sqrt{\pi} 2 ^{inf ty} _{k =1} \left( 1- (1)/(2 k) \right) ^{k( -1 ) ^{k}}

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

The following two representations for

\Gamma\left(\tfrac 34 \right) were given by I. Mező\sqrt{ (\pi\sqrt{e ^{\pi})/() }}\frac=i\sum_^\infty e^\theta_1\left(\frac(2k-1),e^\right),

and

\sqrt{ (\pi)/(2) }\frac=\sum_^\infty\frac,

where and are two of the Jacobi theta functions.

Certain values of the gamma function can also be written in terms of the hypergeometric function. For instance,

\Gamma\left( (1)/(4) \right) ^{4} = (32 \pi ^{3})/(\sqrt{ 33) } _F_\left(\frac,\ \frac,\ \frac;\ 1,\ 1;\ \frac\right)

and

\Gamma\left( (1)/(3) \right) ^{6} = (12 \pi ^{4})/(\sqrt{ 10) } _F_\left(\frac,\ \frac,\ \frac;\ 1,\ 1;\ -\frac\right)

however it is an open question whether this is possible for all rational inputs to the gamma function.^[3]

Products

Some product identities include:

\prod ^{2} _{r =1} \Gamma\left(\tfrac{r}{ 3 }\right) = (2 \pi)/(\sqrt 3) ≈ 3 . 6275987284684357012 \prod ^{3} _{r =1} \Gamma\left(\tfrac{r}{ 4 }\right) = \sqrt{ 2 \pi ^{3}} ≈ 7 . 8748049728612098721 \prod ^{4} _{r =1} \Gamma\left(\tfrac{r}{ 5 }\right) = (4 \pi ^{2})/(\sqrt 5) ≈ 17 . 6552850814935242483 \prod ^{5} _{r =1} \Gamma\left(\tfrac{r}{ 6 }\right) =4 \sqrt{ (\pi ^{5})/(3 }) ≈ 40 . 3993191220037900785 \prod ^{6} _{r =1} \Gamma\left(\tfrac{r}{ 7 }\right) = (8 \pi ^{3})/(\sqrt 7) ≈ 93 . 7541682035825037970 \prod ^{7} _{r =1} \Gamma\left(\tfrac{r}{ 8 }\right) = 4 \sqrt{\pi ^{7}} ≈ 219 . 8287780169572636207

In general:

\prod ^{n} _{r =1} \Gamma\left(\tfrac{r}{n +1 }\right) = \sqrt{ (( 2 \pi) ^{n})/(n +1) }

Other rational relations include

(\Gamma\left(\tfrac 15 \right)\Gamma\left(\tfrac{ 4)/(15) \right)}{\Gamma\left(\tfrac 13 \right)\Gamma\left(\tfrac{ 2 }{ 15 }\right)} = (\sqrt{ 2)/(+ \sqrt{ 6- \sqrt[ 20 ]{ 3 }}{\sqrt[ 6 ]{ 5 }\sqrt[ 4 ]{ 5- (7)/(\sqrt 5)) (6)/(\sqrt 5) }}}(\Gamma\left(\tfrac{ 1)/(20) \right)\Gamma\left(\tfrac{ 9 }{ 20 }\right)}{\Gamma\left(\tfrac{ 3 }{ 20 }\right)\Gamma\left(\tfrac{ 7 }{ 20 }\right)} = (\sqrt[ 4 ]{ 5)/(\left( 1+ \sqrt{ 5 }\right)}{ 2 }) (\Gamma\left( (1)/(\right) ^{2} 5))/(\Gamma\left( (1)/(\right)\Gamma\left( (3)/(10) \right) 10)) = (\sqrt{ 1+ \sqrt{ 5)/() }}

and many more relations for

\Gamma\left(\tfrac{n}{d}\right) where the denominator d divides 24 or 60.^[4]

Imaginary and complex arguments

The gamma function at the imaginary unit gives, :

\Gamma(i) = ( -1+ i)! ≈ -0 . 1549 - 0 . 4980 i. It may also be given in terms of the Barnes -function:\Gamma(i) = (G( 1+ i))/(G(i)) = e ^{-log} .

Because of the Euler Reflection Formula, and the fact that

\Gamma(\overline{z}) = \overline{\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= (\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) ≈ 0 . 498 - 0 . 155 i \Gamma( 1 - i) = - i\Gamma( - i) ≈ 0 . 498 + 0 . 155 i \Gamma(\tfrac 12 + \tfrac 12 i) ≈ 0 . 8181639995 - 0 . 7633138287 i \Gamma(\tfrac 12 - \tfrac 12 i) ≈ 0 . 8181639995 + 0 . 7633138287 i \Gamma( 5 + 3 i) ≈ 0 . 0160418827 - 9 . 4332932898 i \Gamma( 5 - 3 i) ≈ 0 . 0160418827 + 9 . 4332932898 i.

Other constants

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

x _{min} = 1 . 461632144968362341262 \ldots

with the value

\Gamma\left(x _{min} \right) = 0 . 885603194410888 \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:

See also

• Chowla–Selberg formula

Further reading

• V. S.. Adamchik. Multiple Gamma Function and Its Application to Computation of Series. The Ramanujan Journal. 9. 3. 10.1007/s11139-005-1868-3. 2005. 271 - 288. 2173489. math/0308074. 15670340.
• W. . Duke. Ö. . Imamoglu. Special values of multiple gamma functions. Journal de Théorie des Nombres de Bordeaux. 18. 1. 2006. 2245878. 113 - 123. 10.5802/jtnb.536.

Notes and References

1. Waldschmidt . Michel . 2006 . Transcendence of periods: the state of the art . Pure and Applied Mathematics Quarterly . 2 . 2 . 435–463.
2. J. M.. Borwein. I. J.. Zucker. Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind. IMA Journal of Numerical Analysis. 12. 4. 519 - 526. 1992. 1186733. 10.1093/imanum/12.4.519.
3. Johansson, F. (2023). Arbitrary-precision computation of the gamma function. Maple Transactions, 3(1).
4. Vidūnas . Raimundas . math/0403510 . 10.2206/kyushujm.59.267 . 2 . Kyushu Journal of Mathematics . 2188592 . 267–283 . Expressions for values of the gamma function . 59 . 2005.