Particular values of the gamma function

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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)!,} {\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}}} {\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}}} {\displaystyle {\frac {k}{2}}} where k ∈ 2 N ∗ + 1 {\displaystyle k\in 2\mathbb {N} ^{*}+1} {\displaystyle k\in 2\mathbb {N} ^{*}+1} is an odd integer greater than or equal to 3 {\displaystyle 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}}}}\,,} {\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}}} {\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,

Γ ( 1 2 ) {\displaystyle \Gamma \left({\tfrac {1}{2}}\right)\,} {\displaystyle \Gamma \left({\tfrac {1}{2}}\right)\,} = π {\displaystyle ={\sqrt {\pi }}\,} {\displaystyle ={\sqrt {\pi }}\,} ≈ 1.772 453 850 905 516 0273 , {\displaystyle \approx 1.772\,453\,850\,905\,516\,0273\,,} {\displaystyle \approx 1.772\,453\,850\,905\,516\,0273\,,} OEIS: A002161
Γ ( 3 2 ) {\displaystyle \Gamma \left({\tfrac {3}{2}}\right)\,} {\displaystyle \Gamma \left({\tfrac {3}{2}}\right)\,} = 1 2 π {\displaystyle ={\tfrac {1}{2}}{\sqrt {\pi }}\,} {\displaystyle ={\tfrac {1}{2}}{\sqrt {\pi }}\,} ≈ 0.886 226 925 452 758 0137 , {\displaystyle \approx 0.886\,226\,925\,452\,758\,0137\,,} {\displaystyle \approx 0.886\,226\,925\,452\,758\,0137\,,} OEIS: A019704
Γ ( 5 2 ) {\displaystyle \Gamma \left({\tfrac {5}{2}}\right)\,} {\displaystyle \Gamma \left({\tfrac {5}{2}}\right)\,} = 3 4 π {\displaystyle ={\tfrac {3}{4}}{\sqrt {\pi }}\,} {\displaystyle ={\tfrac {3}{4}}{\sqrt {\pi }}\,} ≈ 1.329 340 388 179 137 0205 , {\displaystyle \approx 1.329\,340\,388\,179\,137\,0205\,,} {\displaystyle \approx 1.329\,340\,388\,179\,137\,0205\,,} OEIS: A245884
Γ ( 7 2 ) {\displaystyle \Gamma \left({\tfrac {7}{2}}\right)\,} {\displaystyle \Gamma \left({\tfrac {7}{2}}\right)\,} = 15 8 π {\displaystyle ={\tfrac {15}{8}}{\sqrt {\pi }}\,} {\displaystyle ={\tfrac {15}{8}}{\sqrt {\pi }}\,} ≈ 3.323 350 970 447 842 5512 , {\displaystyle \approx 3.323\,350\,970\,447\,842\,5512\,,} {\displaystyle \approx 3.323\,350\,970\,447\,842\,5512\,,} OEIS: A245885

and by means of the reflection formula,

Γ ( − 1 2 ) {\displaystyle \Gamma \left(-{\tfrac {1}{2}}\right)\,} {\displaystyle \Gamma \left(-{\tfrac {1}{2}}\right)\,} = − 2 π {\displaystyle =-2{\sqrt {\pi }}\,} {\displaystyle =-2{\sqrt {\pi }}\,} ≈ − 3.544 907 701 811 032 0546 , {\displaystyle \approx -3.544\,907\,701\,811\,032\,0546\,,} {\displaystyle \approx -3.544\,907\,701\,811\,032\,0546\,,} OEIS: A019707
Γ ( − 3 2 ) {\displaystyle \Gamma \left(-{\tfrac {3}{2}}\right)\,} {\displaystyle \Gamma \left(-{\tfrac {3}{2}}\right)\,} = 4 3 π {\displaystyle ={\tfrac {4}{3}}{\sqrt {\pi }}\,} {\displaystyle ={\tfrac {4}{3}}{\sqrt {\pi }}\,} ≈ 2.363 271 801 207 354 7031 , {\displaystyle \approx 2.363\,271\,801\,207\,354\,7031\,,} {\displaystyle \approx 2.363\,271\,801\,207\,354\,7031\,,} OEIS: A245886
Γ ( − 5 2 ) {\displaystyle \Gamma \left(-{\tfrac {5}{2}}\right)\,} {\displaystyle \Gamma \left(-{\tfrac {5}{2}}\right)\,} = − 8 15 π {\displaystyle =-{\tfrac {8}{15}}{\sqrt {\pi }}\,} {\displaystyle =-{\tfrac {8}{15}}{\sqrt {\pi }}\,} ≈ − 0.945 308 720 482 941 8812 , {\displaystyle \approx -0.945\,308\,720\,482\,941\,8812\,,} {\displaystyle \approx -0.945\,308\,720\,482\,941\,8812\,,} OEIS: A245887

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}}} {\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} {\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} {\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} {\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} {\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} {\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} {\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 } {\displaystyle \lim _{n\to \infty }\left(n-\Gamma \left({\tfrac {1}{n}}\right)\right)=\gamma }

where γ {\displaystyle \gamma } {\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} {\displaystyle n\geq 2} at least one of the two numbers  Γ ( 1 n ) {\displaystyle \Gamma \left({\tfrac {1}{n}}\right)} {\displaystyle \Gamma \left({\tfrac {1}{n}}\right)} and Γ ( 2 n ) {\displaystyle \Gamma \left({\tfrac {2}{n}}\right)} {\displaystyle \Gamma \left({\tfrac {2}{n}}\right)} is transcendental.[1]

The number Γ ( 1 4 ) {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)} {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)} is related to the lemniscate constant ϖ {\displaystyle \varpi } {\displaystyle \varpi } by

Γ ( 1 4 ) = 2 ϖ 2 π {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)={\sqrt {2\varpi {\sqrt {2\pi }}}}} {\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}}} {\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[2]

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}}}}} {\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)}}}} {\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}}}}.} {\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)} {\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}}} {\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[3]

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),} {\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}}}},} {\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:[4]

∫ 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)} {\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 ⁡ 3 − 12 Γ ( 1 3 ) ) {\displaystyle \int _{1}^{\infty }{\frac {\ln \ln t}{1+t+t^{2}}}={\frac {\pi }{6{\sqrt {3}}}}\left(8\ln 2\pi -3\ln 3-12\Gamma \left({\tfrac {1}{3}}\right)\right)} {\displaystyle \int _{1}^{\infty }{\frac {\ln \ln t}{1+t+t^{2}}}={\frac {\pi }{6{\sqrt {3}}}}\left(8\ln 2\pi -3\ln 3-12\Gamma \left({\tfrac {1}{3}}\right)\right)}

One also has (see[5] and [6]):

Ai ⁡ ( 0 ) = 1 3 2 / 3 Γ ( 2 3 ) {\displaystyle \operatorname {Ai} (0)={\frac {1}{3^{2/3}\,\Gamma \!\left({\frac {2}{3}}\right)}}} {\displaystyle \operatorname {Ai} (0)={\frac {1}{3^{2/3}\,\Gamma \!\left({\frac {2}{3}}\right)}}}, Bi ⁡ ( 0 ) = 1 3 1 / 6 Γ ( 2 3 ) {\displaystyle \operatorname {Bi} (0)={\frac {1}{3^{1/6}\,\Gamma \!\left({\frac {2}{3}}\right)}}} {\displaystyle \operatorname {Bi} (0)={\frac {1}{3^{1/6}\,\Gamma \!\left({\frac {2}{3}}\right)}}} ζ A i ( 2 ) = 3 5 / 3 Γ 4 ( 2 3 ) 4 π 2 {\displaystyle \zeta _{\mathrm {Ai} }(2)={\frac {3^{5/3}\Gamma ^{4}({\frac {2}{3}})}{4\pi ^{2}}}} {\displaystyle \zeta _{\mathrm {Ai} }(2)={\frac {3^{5/3}\Gamma ^{4}({\frac {2}{3}})}{4\pi ^{2}}}}

where Ai(x), Bi(x) are Airy functions and ζ A i ( x ) {\displaystyle \zeta _{\mathrm {Ai} }(x)} {\displaystyle \zeta _{\mathrm {Ai} }(x)} denotes the Airy zeta function.

Products

Some product identities include:

∏ r = 1 2 Γ ( r 3 ) = 2 π 3 ≈ 3.627 598 728 468 435 7012 {\displaystyle \prod _{r=1}^{2}\Gamma \left({\tfrac {r}{3}}\right)={\frac {2\pi }{\sqrt {3}}}\approx 3.627\,598\,728\,468\,435\,7012} {\displaystyle \prod _{r=1}^{2}\Gamma \left({\tfrac {r}{3}}\right)={\frac {2\pi }{\sqrt {3}}}\approx 3.627\,598\,728\,468\,435\,7012} OEIS: A186706
∏ r = 1 3 Γ ( r 4 ) = 2 π 3 ≈ 7.874 804 972 861 209 8721 {\displaystyle \prod _{r=1}^{3}\Gamma \left({\tfrac {r}{4}}\right)={\sqrt {2\pi ^{3}}}\approx 7.874\,804\,972\,861\,209\,8721} {\displaystyle \prod _{r=1}^{3}\Gamma \left({\tfrac {r}{4}}\right)={\sqrt {2\pi ^{3}}}\approx 7.874\,804\,972\,861\,209\,8721} OEIS: A220610
∏ r = 1 4 Γ ( r 5 ) = 4 π 2 5 ≈ 17.655 285 081 493 524 2483 {\displaystyle \prod _{r=1}^{4}\Gamma \left({\tfrac {r}{5}}\right)={\frac {4\pi ^{2}}{\sqrt {5}}}\approx 17.655\,285\,081\,493\,524\,2483} {\displaystyle \prod _{r=1}^{4}\Gamma \left({\tfrac {r}{5}}\right)={\frac {4\pi ^{2}}{\sqrt {5}}}\approx 17.655\,285\,081\,493\,524\,2483}
∏ r = 1 5 Γ ( r 6 ) = 4 π 5 3 ≈ 40.399 319 122 003 790 0785 {\displaystyle \prod _{r=1}^{5}\Gamma \left({\tfrac {r}{6}}\right)=4{\sqrt {\frac {\pi ^{5}}{3}}}\approx 40.399\,319\,122\,003\,790\,0785} {\displaystyle \prod _{r=1}^{5}\Gamma \left({\tfrac {r}{6}}\right)=4{\sqrt {\frac {\pi ^{5}}{3}}}\approx 40.399\,319\,122\,003\,790\,0785}
∏ r = 1 6 Γ ( r 7 ) = 8 π 3 7 ≈ 93.754 168 203 582 503 7970 {\displaystyle \prod _{r=1}^{6}\Gamma \left({\tfrac {r}{7}}\right)={\frac {8\pi ^{3}}{\sqrt {7}}}\approx 93.754\,168\,203\,582\,503\,7970} {\displaystyle \prod _{r=1}^{6}\Gamma \left({\tfrac {r}{7}}\right)={\frac {8\pi ^{3}}{\sqrt {7}}}\approx 93.754\,168\,203\,582\,503\,7970}
∏ r = 1 7 Γ ( r 8 ) = 4 π 7 ≈ 219.828 778 016 957 263 6207 {\displaystyle \prod _{r=1}^{7}\Gamma \left({\tfrac {r}{8}}\right)=4{\sqrt {\pi ^{7}}}\approx 219.828\,778\,016\,957\,263\,6207} {\displaystyle \prod _{r=1}^{7}\Gamma \left({\tfrac {r}{8}}\right)=4{\sqrt {\pi ^{7}}}\approx 219.828\,778\,016\,957\,263\,6207}

In general:

∏ r = 1 n Γ ( r n + 1 ) = ( 2 π ) n n + 1 {\displaystyle \prod _{r=1}^{n}\Gamma \left({\tfrac {r}{n+1}}\right)={\sqrt {\frac {(2\pi )^{n}}{n+1}}}} {\displaystyle \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 ∏ r = 1 3 Γ ( r 4 ) {\displaystyle \prod _{r=1}^{3}\Gamma \left({\tfrac {r}{4}}\right)} {\displaystyle \prod _{r=1}^{3}\Gamma \left({\tfrac {r}{4}}\right)}, Γ ( 1 4 ) {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)} {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)} and Γ ( 2 4 ) {\displaystyle \Gamma \left({\tfrac {2}{4}}\right)} {\displaystyle \Gamma \left({\tfrac {2}{4}}\right)}, can be deduced:

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

Γ ( 1 5 ) Γ ( 4 15 ) Γ ( 1 3 ) Γ ( 2 15 ) = 2 3 20 5 6 5 − 7 5 + 6 − 6 5 4 {\displaystyle {\frac {\Gamma \left({\tfrac {1}{5}}\right)\Gamma \left({\tfrac {4}{15}}\right)}{\Gamma \left({\tfrac {1}{3}}\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}}}}}}}}}} {\displaystyle {\frac {\Gamma \left({\tfrac {1}{5}}\right)\Gamma \left({\tfrac {4}{15}}\right)}{\Gamma \left({\tfrac {1}{3}}\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}}}}}}}}}}
Γ ( 1 20 ) Γ ( 9 20 ) Γ ( 3 20 ) Γ ( 7 20 ) = 5 4 ( 1 + 5 ) 2 {\displaystyle {\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}}} {\displaystyle {\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}}}[7]
Γ ( 1 5 ) 2 Γ ( 1 10 ) Γ ( 3 10 ) = 1 + 5 2 7 10 5 4 {\displaystyle {\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}}}}} {\displaystyle {\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(⁠n/d⁠) where the denominator d divides 24 or 60.[8]

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:

Γ ( 11 42 ) Γ ( 2 7 ) Γ ( 1 21 ) Γ ( 1 2 ) = 8 sin ⁡ ( π 7 ) sin ⁡ ( π 21 ) sin ⁡ ( 4 π 21 ) sin ⁡ ( 5 π 21 ) 2 1 42 3 9 28 7 1 3 {\displaystyle {\frac {\Gamma \left({\frac {11}{42}}\right)\Gamma \left({\frac {2}{7}}\right)}{\Gamma \left({\frac {1}{21}}\right)\Gamma \left({\frac {1}{2}}\right)}}={\frac {8\sin \left({\frac {\pi }{7}}\right){\sqrt {\sin \left({\frac {\pi }{21}}\right)\sin \left({\frac {4\pi }{21}}\right)\sin \left({\frac {5\pi }{21}}\right)}}}{2^{\frac {1}{42}}3^{\frac {9}{28}}7^{\frac {1}{3}}}}} {\displaystyle {\frac {\Gamma \left({\frac {11}{42}}\right)\Gamma \left({\frac {2}{7}}\right)}{\Gamma \left({\frac {1}{21}}\right)\Gamma \left({\frac {1}{2}}\right)}}={\frac {8\sin \left({\frac {\pi }{7}}\right){\sqrt {\sin \left({\frac {\pi }{21}}\right)\sin \left({\frac {4\pi }{21}}\right)\sin \left({\frac {5\pi }{21}}\right)}}}{2^{\frac {1}{42}}3^{\frac {9}{28}}7^{\frac {1}{3}}}}}[9]

Imaginary and complex arguments

The gamma function at the imaginary unit i = √−1 gives OEIS: A212877, OEIS: A212878:

Γ ( i ) = ( − 1 + i ) ! ≈ − 0.1549 − 0.4980 i . {\displaystyle \Gamma (i)=(-1+i)!\approx -0.1549-0.4980i.} {\displaystyle \Gamma (i)=(-1+i)!\approx -0.1549-0.4980i.}

It may also be given in terms of the Barnes G-function:

Γ ( i ) = G ( 1 + i ) G ( i ) = e − log ⁡ G ( i ) + log ⁡ G ( 1 + i ) . {\displaystyle \Gamma (i)={\frac {G(1+i)}{G(i)}}=e^{-\log G(i)+\log G(1+i)}.} {\displaystyle \Gamma (i)={\frac {G(1+i)}{G(i)}}=e^{-\log G(i)+\log G(1+i)}.}

Γ ( i ) {\displaystyle \Gamma (i)} {\displaystyle \Gamma (i)} appears in the below integral evaluation:[10]

∫ 0 π / 2 { cot ⁡ ( x ) } d x = 1 − π 2 + i 2 log ⁡ ( π sinh ⁡ ( π ) Γ ( i ) 2 ) . {\displaystyle \int _{0}^{\pi /2}\{\cot(x)\}\,dx=1-{\frac {\pi }{2}}+{\frac {i}{2}}\log \left({\frac {\pi }{\sinh(\pi )\Gamma (i)^{2}}}\right).} {\displaystyle \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 { ⋅ } {\displaystyle \{\cdot \}} {\displaystyle \{\cdot \}} denotes the fractional part.

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

| Γ ( i κ ) | 2 = π κ sinh ⁡ ( π κ ) {\displaystyle \left|\Gamma (i\kappa )\right|^{2}={\frac {\pi }{\kappa \sinh(\pi \kappa )}}} {\displaystyle \left|\Gamma (i\kappa )\right|^{2}={\frac {\pi }{\kappa \sinh(\pi \kappa )}}}

The above integral therefore relates to the phase of Γ ( i ) {\displaystyle \Gamma (i)} {\displaystyle \Gamma (i)}.

The gamma function with other complex arguments returns

Γ ( 1 + i ) = i Γ ( i ) ≈ 0.498 − 0.155 i {\displaystyle \Gamma (1+i)=i\Gamma (i)\approx 0.498-0.155i} {\displaystyle \Gamma (1+i)=i\Gamma (i)\approx 0.498-0.155i}
Γ ( 1 − i ) = − i Γ ( − i ) ≈ 0.498 + 0.155 i {\displaystyle \Gamma (1-i)=-i\Gamma (-i)\approx 0.498+0.155i} {\displaystyle \Gamma (1-i)=-i\Gamma (-i)\approx 0.498+0.155i}
Γ ( 1 2 + 1 2 i ) ≈ 0.818 163 9995 − 0.763 313 8287 i {\displaystyle \Gamma ({\tfrac {1}{2}}+{\tfrac {1}{2}}i)\approx 0.818\,163\,9995-0.763\,313\,8287\,i} {\displaystyle \Gamma ({\tfrac {1}{2}}+{\tfrac {1}{2}}i)\approx 0.818\,163\,9995-0.763\,313\,8287\,i}
Γ ( 1 2 − 1 2 i ) ≈ 0.818 163 9995 + 0.763 313 8287 i {\displaystyle \Gamma ({\tfrac {1}{2}}-{\tfrac {1}{2}}i)\approx 0.818\,163\,9995+0.763\,313\,8287\,i} {\displaystyle \Gamma ({\tfrac {1}{2}}-{\tfrac {1}{2}}i)\approx 0.818\,163\,9995+0.763\,313\,8287\,i}
Γ ( 5 + 3 i ) ≈ 0.016 041 8827 − 9.433 293 2898 i {\displaystyle \Gamma (5+3i)\approx 0.016\,041\,8827-9.433\,293\,2898\,i} {\displaystyle \Gamma (5+3i)\approx 0.016\,041\,8827-9.433\,293\,2898\,i}
Γ ( 5 − 3 i ) ≈ 0.016 041 8827 + 9.433 293 2898 i . {\displaystyle \Gamma (5-3i)\approx 0.016\,041\,8827+9.433\,293\,2898\,i.} {\displaystyle \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 … {\displaystyle x_{\min }=1.461\,632\,144\,968\,362\,341\,262\,659\,5423\ldots \,} {\displaystyle x_{\min }=1.461\,632\,144\,968\,362\,341\,262\,659\,5423\ldots \,} OEIS: A030169

with the value

Γ ( x min ) = 0.885 603 194 410 888 700 278 815 9005 … {\displaystyle \Gamma \left(x_{\min }\right)=0.885\,603\,194\,410\,888\,700\,278\,815\,9005\ldots \,} {\displaystyle \Gamma \left(x_{\min }\right)=0.885\,603\,194\,410\,888\,700\,278\,815\,9005\ldots \,} OEIS: A030171.

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:

Approximate local extrema of Γ(x)
xΓ(x)OEIS
−0.5040830082644554092582693045−3.5446436111550050891219639933OEIS: A175472
−1.5734984731623904587782860437−2.3024072583396801358235820396OEIS: A175473
−2.6107208684441446500015377157−0.8881363584012419200955280294OEIS: A175474
−3.6352933664369010978391815669−0.2451275398343662504382300889OEIS: A256681
−4.6532377617431424417145981511−0.0527796395873194007604835708OEIS: A256682
−5.6671624415568855358494741745−0.0093245944826148505217119238OEIS: A256683
−6.6784182130734267428298558886−0.0013973966089497673013074887OEIS: A256684
−7.6877883250316260374400988918−0.0001818784449094041881014174OEIS: A256685
−8.6957641638164012664887761608−0.0000209252904465266687536973OEIS: A256686
−9.7026725400018637360844267649−0.0000021574161045228505405031OEIS: A256687

The only values of x > 0 for which Γ(x) = x are x = 1 and x ≈ 3.5623822853908976914156443427... OEIS: A218802.

See also

References

  1. Waldschmidt, Michel (2006). "Transcendence of periods: the state of the art". Pure and Applied Mathematics Quarterly. 2 (2): 435–463. doi:10.4310/PAMQ.2006.v2.n2.a3.
  2. "Archived copy". Retrieved 2015-03-09.
  3. Mező, István (2013), "Duplication formulae involving Jacobi theta functions and Gosper's q-trigonometric functions", Proceedings of the American Mathematical Society, 141 (7): 2401–2410, doi:10.1090/s0002-9939-2013-11576-5
  4. Blagouchine, Iaroslav V. (2014-10-01). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results". The Ramanujan Journal. 35 (1): 21–110. doi:10.1007/s11139-013-9528-5. ISSN 1572-9303.
  5. "Airy and Related Function". dlmf.nist.gov. Retrieved 9 October 2022.
  6. Weisstein, Eric W. "Airy Zeta Function". MathWorld.
  7. Weisstein, Eric W. "Gamma Function". MathWorld.
  8. Raimundas Vidūnas, Expressions for Values of the Gamma Function
  9. math.stackexchange.com
  10. The webpage of István Mező

Further reading