The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer, half-integer, and some other rational 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,
Γ ( n ) = ( n − 1 ) ! , {\displaystyle \Gamma (n)=(n-1)!,}
and hence
Γ ( 1 ) = 1 , Γ ( 2 ) = 1 , Γ ( 3 ) = 2 , Γ ( 4 ) = 6 , Γ ( 5 ) = 24 , {\displaystyle {\begin{aligned}\Gamma (1)&=1,\\\Gamma (2)&=1,\\\Gamma (3)&=2,\\\Gamma (4)&=6,\\\Gamma (5)&=24,\end{aligned}}}
and so on. For non-positive integers, the gamma function is not defined. For positive half-integers k 2 {\displaystyle {\frac {k}{2}}} where k ∈ 2 N ∗ + 1 {\displaystyle k\in 2\mathbb {N} ^{*}+1} is an odd integer greater than or equal to 3 {\displaystyle 3} , the function values are given exactly by
Γ ( k 2 ) = π ( k − 2 ) ! ! 2 k − 1 2 , {\displaystyle \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:
Γ ( 1 2 + n ) = ( 2 n − 1 ) ! ! 2 n π = ( 2 n ) ! 4 n n ! π Γ ( 1 2 − n ) = ( − 2 ) n ( 2 n − 1 ) ! ! π = ( − 4 ) n n ! ( 2 n ) ! π {\displaystyle {\begin{aligned}\Gamma \left({\tfrac {1}{2}}+n\right)&={\frac {(2n-1)!!}{2^{n}}}\,{\sqrt {\pi }}={\frac {(2n)!}{4^{n}n!}}{\sqrt {\pi }}\\\Gamma \left({\tfrac {1}{2}}-n\right)&={\frac {(-2)^{n}}{(2n-1)!!}}\,{\sqrt {\pi }}={\frac {(-4)^{n}n!}{(2n)!}}{\sqrt {\pi }}\end{aligned}}}
where n!! denotes the double factorial. In particular,
and by means of the reflection formula,
General rational argument In analogy with the half-integer formula,
Γ ( n + 1 3 ) = Γ ( 1 3 ) ( 3 n − 2 ) ! ! ! 3 n Γ ( n + 1 4 ) = Γ ( 1 4 ) ( 4 n − 3 ) ! ! ! ! 4 n Γ ( n + 1 q ) = Γ ( 1 q ) ( q n − ( q − 1 ) ) ! ( q ) q n Γ ( n + p q ) = Γ ( p q ) 1 q n ∏ k = 1 n ( k q + p − q ) {\displaystyle {\begin{aligned}\Gamma \left(n+{\tfrac {1}{3}}\right)&=\Gamma \left({\tfrac {1}{3}}\right){\frac {(3n-2)!!!}{3^{n}}}\\\Gamma \left(n+{\tfrac {1}{4}}\right)&=\Gamma \left({\tfrac {1}{4}}\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}(kq+p-q)\end{aligned}}}
where n!(q) denotes the qth multifactorial of n. Numerically,
Γ ( 1 3 ) ≈ 2.678 938 534 707 747 6337 {\displaystyle \Gamma \left({\tfrac {1}{3}}\right)\approx 2.678\,938\,534\,707\,747\,6337} OEIS: A073005
Γ ( 1 4 ) ≈ 3.625 609 908 221 908 3119 {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)\approx 3.625\,609\,908\,221\,908\,3119} OEIS: A068466
Γ ( 1 5 ) ≈ 4.590 843 711 998 803 0532 {\displaystyle \Gamma \left({\tfrac {1}{5}}\right)\approx 4.590\,843\,711\,998\,803\,0532} OEIS: A175380
Γ ( 1 6 ) ≈ 5.566 316 001 780 235 2043 {\displaystyle \Gamma \left({\tfrac {1}{6}}\right)\approx 5.566\,316\,001\,780\,235\,2043} OEIS: A175379
Γ ( 1 7 ) ≈ 6.548 062 940 247 824 4377 {\displaystyle \Gamma \left({\tfrac {1}{7}}\right)\approx 6.548\,062\,940\,247\,824\,4377} OEIS: A220086
Γ ( 1 8 ) ≈ 7.533 941 598 797 611 9047 {\displaystyle \Gamma \left({\tfrac {1}{8}}\right)\approx 7.533\,941\,598\,797\,611\,9047} OEIS: A203142. Additionally,
lim n → ∞ ( n − Γ ( 1 n ) ) = γ {\displaystyle \lim _{n\to \infty }\left(n-\Gamma \left({\tfrac {1}{n}}\right)\right)=\gamma }
where γ {\displaystyle \gamma } is the Euler–Mascheroni constant. It is unknown whether these constants are transcendental in general, but Γ(1/3) and Γ(1/4) were shown to be transcendental by G. V. Chudnovsky. Γ(1/4) / 4√π has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that Γ(1/4), π, and eπ are algebraically independent. For n ≥ 2 {\displaystyle n\geq 2} at least one of the two numbers Γ ( 1 n ) {\displaystyle \Gamma \left({\tfrac {1}{n}}\right)} and Γ ( 2 n ) {\displaystyle \Gamma \left({\tfrac {2}{n}}\right)} is transcendental. The number Γ ( 1 4 ) {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)} is related to the lemniscate constant ϖ {\displaystyle \varpi } by
Γ ( 1 4 ) = 2 ϖ 2 π {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)={\sqrt {2\varpi {\sqrt {2\pi }}}}}
Borwein and Zucker have found that Γ(n/24) 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:
Γ ( 1 6 ) = 3 π Γ ( 1 3 ) 2 2 3 Γ ( 1 4 ) = 2 K ( 1 2 ) π Γ ( 1 3 ) = 2 7 / 9 π K ( 3 − 1 2 2 ) 3 3 12 Γ ( 1 8 ) Γ ( 3 8 ) = 8 2 4 ( 2 − 1 ) π K ( 3 − 2 2 ) Γ ( 1 8 ) Γ ( 3 8 ) = 2 ( 1 + 2 ) K ( 1 2 ) π 4 {\displaystyle {\begin{aligned}\Gamma \left({\tfrac {1}{6}}\right)&={\frac {{\sqrt {\frac {3}{\pi }}}\Gamma \left({\frac {1}{3}}\right)^{2}}{\sqrt[{3}]{2}}}\\\Gamma \left({\tfrac {1}{4}}\right)&=2{\sqrt {K\left({\tfrac {1}{\sqrt {2}}}\right){\sqrt {\pi }}}}\\\Gamma \left({\tfrac {1}{3}}\right)&={\frac {2^{7/9}{\sqrt[{3}]{\pi K\left({\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}\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{aligned}}}
No similar relations are known for Γ(1/5) or other denominators. In particular, where AGM() is the arithmetic–geometric mean, we have
Γ ( 1 3 ) = 2 1 9 ( 2 π ) 2 3 3 1 12 ⋅ AGM ( 2 , 2 + 3 ) 1 3 {\displaystyle \Gamma \left({\tfrac {1}{3}}\right)={\frac {2^{\frac {1}{9}}(2\pi )^{\frac {2}{3}}}{3^{\frac {1}{12}}\cdot \operatorname {AGM} \left(2,{\sqrt {2+{\sqrt {3}}}}\right)^{\frac {1}{3}}}}}
Γ ( 1 4 ) = ( 2 π ) 3 2 AGM ( 2 , 1 ) {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)={\sqrt {\frac {(2\pi )^{\frac {3}{2}}}{\operatorname {AGM} \left({\sqrt {2}},1\right)}}}}
Γ ( 1 6 ) = 2 14 9 ⋅ 3 1 3 ⋅ π 5 6 AGM ( 1 + 3 , 8 ) 2 3 . {\displaystyle \Gamma \left({\tfrac {1}{6}}\right)={\frac {2^{\frac {14}{9}}\cdot 3^{\frac {1}{3}}\cdot \pi ^{\frac {5}{6}}}{\operatorname {AGM} \left(1+{\sqrt {3}},{\sqrt {8}}\right)^{\frac {2}{3}}}}.}
Other formulas include the infinite products
Γ ( 1 4 ) = ( 2 π ) 3 4 ∏ k = 1 ∞ tanh ( π k 2 ) {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)=(2\pi )^{\frac {3}{4}}\prod _{k=1}^{\infty }\tanh \left({\frac {\pi k}{2}}\right)}
and
Γ ( 1 4 ) = A 3 e − G π 2 1 6 π ∏ k = 1 ∞ ( 1 − 1 2 k ) k ( − 1 ) k {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)=A^{3}e^{-{\frac {G}{\pi }}}2^{\frac {1}{6}}{\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(3/4) were given by I. Mező
π e π 2 1 Γ ( 3 4 ) 2 = i ∑ k = − ∞ ∞ e π ( k − 2 k 2 ) θ 1 ( i π 2 ( 2 k − 1 ) , e − π ) , {\displaystyle {\sqrt {\frac {\pi {\sqrt {e^{\pi }}}}{2}}}{\frac {1}{\Gamma \left({\frac {3}{4}}\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
π 2 1 Γ ( 3 4 ) 2 = ∑ k = − ∞ ∞ θ 4 ( i k π , e − π ) e 2 π k 2 , {\displaystyle {\sqrt {\frac {\pi }{2}}}{\frac {1}{\Gamma \left({\frac {3}{4}}\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:
∫ 1 ∞ ln ln t 1 + t 2 = π 4 ( 2 ln 2 + 3 ln π − 4 Γ ( 1 4 ) ) {\displaystyle \int _{1}^{\infty }{\frac {\ln \ln t}{1+t^{2}}}={\frac {\pi }{4}}\left(2\ln 2+3\ln \pi -4\Gamma \left({\tfrac {1}{4}}\right)\right)}
∫ 1 ∞ ln ln t 1 + t + t 2 = π 6 3 ( 8 ln 2 π − 3 ln